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This article may be downloaded for personal use only. Any other use requires
prior permission of the author and AIP Publishing. This article appeared in
Hunt et al, J. Renewable Sustainable Energy 12, 054501 (2020) and may be found
at https://doi.org/10.1063/5.0016753.

# Effect of Aspect Ratio on Cross-Flow Turbine Performance

Aidan Hunt<EMAIL_ADDRESS>Department of Mechanical Engineering, University of
Washington  Carl Stringer  Department of Mechanical Engineering, University of
Washington  Brian Polagye  Department of Mechanical Engineering, University of
Washington

(6 August 2020)

###### Abstract

Cross-flow turbines convert kinetic power in wind or water currents to
mechanical power. Unlike axial-flow turbines, the influence of geometric
parameters on turbine performance is not well-understood, in part because
there are neither generalized analytical formulations nor inexpensive,
accurate numerical models that describe their fluid dynamics. Here, we
experimentally investigate the effect of aspect ratio – the ratio of the blade
span to rotor diameter – on the performance of a straight-bladed cross-flow
turbine in a water channel. To isolate the effect of aspect ratio, all other
non-dimensional parameters are held constant, including the relative
confinement, Froude number, and Reynolds number. The coefficient of
performance is found to be invariant for the range of aspect ratios tested
(0.95 – 1.63), which we ascribe to minimal blade-support interactions for this
turbine design. Finally, a subset of experiments is repeated without
controlling for the Froude number and the coefficient of performance is found
to increase, a consequence of Froude number variation that could mistakenly be
ascribed to aspect ratio. This highlights the importance of rigorous
experimental design when exploring the effect of geometric parameters on
cross-flow turbine performance.

††preprint: JRSE20-AR-00951

##  I Introduction

Cross-flow turbines convert the kinetic power in wind or water currents to
useful mechanical power (Khan  _et al._ , 2009; Dabiri, 2011). While designs
with fixed blade pitch have only a single degree of rotational freedom, static
geometric parameters have important implications for turbine performance.
These parameters include the dimensions of the rotor (e.g., aspect ratio
(Brusca, Lanzafame, and Messina, 2014; Li  _et al._ , 2017), helix angle
(Shiono, Suzuki, and Kiho, 2002), and number of blades (Castelli, De Betta,
and Benini, 2012; Li  _et al._ , 2015)), properties of the blades (e.g., foil
profile (Simão Ferreira and Geurts, 2015; Bedon, De Betta, and Benini, 2016),
preset-pitch angle (Fiedler and Tullis, 2009; Strom, Brunton, and Polagye,
2015), and chord-to-radius ratio (Migliore, Wolfe, and Fanucci, 1980; Balduzzi
_et al._ , 2015)), and the type of support members used to attach the blades
to the central shaft (e.g., struts or end-plates (Strom, Johnson, and Polagye,
2018; Villeneuve, Boudreau, and Dumas, 2019)).

The breadth of this parameter space suggests optimization by reduced-order
models, but this is complicated by the unsteady nature of cross-flow turbine
fluid dynamics. The blade’s angle of attack varies throughout the turbine’s
rotation, potentially inducing dynamic stall (Tsai and Colonius, 2016; Buchner
_et al._ , 2015), and the blades pass disturbed flow during their downstream
sweep, potentially interacting with coherent structures. Such dynamics are
difficult to generalize to the full range of rotor geometries. Unlike blade
element momentum (BEM) models for axial-flow turbines (Burton  _et al._ ,
2011), analytical models (Paraschivoiu, 2002; Islam, Ting, and Fartaj, 2008;
Mohammed  _et al._ , 2019) are often only valid for a specific configuration.
Similarly, the challenges of simulating unsteady boundary layer dynamics (Tsai
and Colonius, 2016; Buchner  _et al._ , 2015), combined with a lack of
exploitable circumferential symmetry and a necessity for high spatial and
temporal resolution (Balduzzi  _et al._ , 2016), make accurate two-dimensional
simulations computationally expensive. Three-dimensional simulations, which
are even more expensive, are required to fully incorporate the effects of
span-wise flow, mixing, and parasitic torque from blade supports (Simão
Ferreira  _et al._ , 2014; Bachant and Wosnik, 2016a; Bianchini  _et al._ ,
2017).

These limitations create a natural gravity towards experimental exploration of
geometric optimization. Here, computational shortcomings are exchanged for
limits on turbine size and flow conditions imposed by the test facility (e.g.,
maximum Reynolds number (Miller  _et al._ , 2018)). Several experimental
studies have investigated the effects of a variety of geometric parameters on
cross-flow turbine performance Shiono, Suzuki, and Kiho (2000, 2002); Fiedler
and Tullis (2009); Li  _et al._ (2015); Strom, Brunton, and Polagye (2015);
Bachant  _et al._ (2016); Strom, Johnson, and Polagye (2018); Strom (2019).
However, experiments must be structured such that the geometric parameter of
interest is fully isolated (i.e., all other non-dimensional parameters that
affect turbine performance must be held constant). This can be difficult to
achieve in practice, and, if neglected, erroneous conclusions may be drawn
about the effect of a geometric parameter.

For example, consider the turbine aspect ratio, a geometric parameter defined
as

$AR=\frac{H}{D},$ (1)

where $H$ is the blade span, and $D$ is the rotor diameter. The aspect ratio
has several practical implications for turbine implementation. First, for a
cross-flow turbine in a cantilevered orientation, as aspect ratio increases
while the diameter is held constant, the moment on the drive train bearings
and supporting structure increases, with attendant cost and design complexity.
Second, as aspect ratio increases, intermediate support structures may be
required between the blades and drive shaft to limit mechanical stress. These
intermediate structures can negatively affect power generation (Strom,
Johnson, and Polagye, 2018).

Previous investigations of the effect of aspect ratio on cross-flow turbine
performance have focused on simulation, although some experiments have been
conducted. Brusca, Lanzafame, and Messina (2014) utilized an analytical model
(multiple stream tube) to characterize the effect of aspect ratios from 0 to 3
on the performance of a straight-bladed vertical-axis wind turbine. The
authors concluded that a decrease in aspect ratio increased efficiency. This
was attributed to an increase in the optimal Reynolds number for lower aspect
ratios, which would be expected to increase turbine performance while in a
Reynolds-dependent regime (Miller  _et al._ , 2018). However, because aspect
ratio was varied by changing the turbine diameter, this simultaneously varied
aspect ratio along with the chord-to-radius ratio $(c/R)$ and solidity
$(Nc/\pi D)$, both of which can significantly affect turbine performance
(Strom, 2019; Shiono, Suzuki, and Kiho, 2000). Specifically, as chord-to-
radius ratio and solidity decrease, the optimal tip-speed ratio increases,
thus increasing the local blade Reynolds number, and overall turbine
efficiency for Reynolds-dependent regimes. In addition, as chord-to-radius
ratio increases, an empirical correction for flow curvature is required
(Migliore, Wolfe, and Fanucci, 1980; Balduzzi  _et al._ , 2015). Consequently,
it would be inappropriate to ascribe the change in performance solely to the
aspect ratio.

Li  _et al._ (2017) investigated the effect of aspect ratio on a straight-
bladed cross-flow wind turbine using a three-dimensional simulation (panel
method). Turbine aspect ratios from 0.4 to 1.2 were investigated, and both the
chord-to-radius ratio and solidity were held constant for all cases. The
simulation predicted an increase in efficiency with increasing aspect ratio.
The authors attributed this to a decrease in the detrimental effects of blade
tip vortices. In other words, as the turbine blade span increased, the tip
losses became relatively less significant.

Hyun  _et al._ (2012) experimentally examined the performance of three-bladed
and four-bladed cross-flow turbines with aspect ratios of 1.5 and 2 in a
circulating water channel with a free surface. As for Li et al., these
experiments held the chord-to-radius ratio and solidity constant. While the
reported turbine efficiency increased with increasing aspect ratio, this
parameter was not fully isolated by the experimental design. The turbines were
tested in a water channel with a constant depth, such that the confinement
varied with blade span: for $AR=1.5$, "blockage" (i.e., the ratio of turbine
projected area to the channel cross-section) was 30%, while for $AR=2$,
blockage was 40%. The authors attributed the improved performance of the
higher aspect ratio turbine to the turbine blades piercing the free surface,
but the higher confinement would increase efficiency (Garrett and Cummins,
2007; Consul, Willden, and McIntosh, 2013; Houlsby and Vogel, 2017; Ross and
Polagye, 2020) and ventilation from piercing the surface (i.e., air entrained
into the rotor) would reduce lift (Young  _et al._ , 2017). Consequently, the
efficiency increase attributed to aspect ratio is inextricably convolved with
confinement and ventilation.

Consequently, to our knowledge, there has been no experimental investigation
that fully isolates the effect of aspect ratio on cross-flow turbine
performance. Further, because of convolved non-dimensional variables in both
simulation and experimental studies, prior work has drawn conflicting
conclusions about the effects of aspect ratio. Here, we evaluate the
performance of a straight-bladed cross-flow turbine with blade-end struts over
a range of aspect ratios through an experimental method that holds all other
relevant non-dimensional parameters constant. In doing so, we provide insight
into the effect of aspect ratio on cross-flow turbine performance and the
experimental design required to fully isolate the influence of geometric
parameters.

##  II Methods

###  II.1 Experimental Set-Up

Figure 1: Schematic of the cross-flow turbine used in this study. Aspect ratio
was varied by changing the blade span, $H$, while holding the diameter, $D$,
constant. The flume width and dynamic depth are given by $w$ and $h$,
respectively. Units are centimeters unless otherwise indicated.

The experimental turbine used in this study was a cross-flow turbine with two
straight blades, as illustrated in Fig. 1. To vary aspect ratio, multiple
turbine blades, each with a different blade span, $H$, were manufactured,
while chord length (4.05 cm), blade profile (NACA 0018), turbine diameter
(17.2 cm), and preset pitch angle ($6^{\circ}$) were held constant. These
parameters correspond to aspect ratios from 0.95 to 1.63 and a constant chord-
to-radius ratio of 0.24. For comparison, the turbines with the smallest and
largest aspect ratios are shown in Fig. 3. The blades were attached to the
central shaft at their ends using thin, hydrodynamic struts (NACA 0008
profile, cstrut = 4.05 cm) to minimize support structure losses and blade-
strut interactions (Strom, Johnson, and Polagye, 2018). Blades and struts were
machined from 6061 aluminum, sanded, coated with a thin layer of grey Dupli-
Color filler primer, and painted black with Krylon ultra-flat black spray
paint.

Experiments were conducted in the Alice C. Tyler recirculating water flume at
the University of Washington. The flume is 75 cm wide and has closed-loop
temperature control for heating or cooling. The experimental setup, shown in
Fig. 3, is as described in Polagye  _et al._ (2019) with the exception of the
bottom load cell, acoustic Doppler velocimeter, and free surface transducer.
Salient details are repeated here.

Figure 2: Comparison of the lowest aspect ratio turbine (left, $AR=0.95$) to
the highest (right, $AR=1.63$).

\begin{overpic}[width=368.57964pt]{Figures/Figure3.png} \put(-1.0,5.4){\small
Suction plate} \put(65.0,9.0){\small Mini40 load cell} \put(11.5,34.75){\small
Turbine} \put(5.25,69.0){\small Shaft coupling}
\put(73.0,74.0){\small\begin{tabular}[]{c}Servomotor,\\\ encoder\end{tabular}}
\put(70.0,85.0){\small Thermal isolation}
\put(-1.0,84.0){\small\begin{tabular}[]{c}Free surface\\\
transducer\end{tabular}} \put(65.5,92.5){\small Mini45 load cell}
\end{overpic} Figure 3: A rendering of the turbine experimental set-up. The
upper load cell is connected to a rigid crossbeam that spans the flume width.

The top end of the turbine shaft is attached via a shaft coupling to a
servomotor (Yaskawa SGMCS-05BC341), which regulates the turbine rotational
rate to a constant, prescribed value. The angular position of the turbine is
measured by the servomotor encoder, and angular velocity is calculated by
differentiation with respect to time. A six-axis load cell (ATI Mini45)
attached to the top of the servomotor measures moments and forces on the
turbine in the horizontal and vertical directions. The bottom end of the
turbine shaft rests in a bearing on a second six-axis load cell (ATI Mini40).
The bearing and load cell are affixed to the bottom of the flume using a
suction plate. Measurements from the encoder and load cells were logged by a
National Instruments PCIe-6351 card through a custom Simulink model in MATLAB
at a sample rate of 1000 Hz.

The inflow velocity was measured using an acoustic Doppler velocimeter (Nortek
Vectrino profiler). A single cell, centered vertically and horizontally in the
water column, was sampled at 64 Hz at a location 5 turbine diameters upstream
from the center shaft. Velocity measurements were de-spiked using the method
of Goring and Nikora (2002). The dynamic depth of the water channel was
measured approximately $0.50D$ upstream and $0.75D$ laterally offset from the
center of the turbine using a free-surface transducer (Omega LVU30 series)
sampling at 10 Hz. The water temperature was measured using a temperature
probe (Omega T-type) with a resolution of $\pm 0.1\,^{\circ}$C and logged
manually every minute during each experiment.

###  II.2 Non-dimensional Parameters

Because the confinement (Garrett and Cummins, 2007; Consul, Willden, and
McIntosh, 2013; Houlsby and Vogel, 2017; Ross and Polagye, 2020), Froude
number (Houlsby and Vogel, 2017; Consul, Willden, and McIntosh, 2013; Gauvin-
Tremblay and Dumas, 2020), and Reynolds number (Bachant and Wosnik, 2016b;
Miller  _et al._ , 2018), can all affect turbine performance, we designed the
experiments to hold these constant. The confinement, or blockage ratio,
experienced by the turbine is defined as the ratio of the turbine’s projected
area to the cross-sectional area of the channel

$\beta=\frac{A_{turbine}}{A_{channel}}=\frac{HD}{hw},$ (2)

where $h$ is the dynamic depth of the water in the flume, and $w$ is the width
of the flume. During testing, each turbine was centered vertically and
horizontally in the water column, and the water depth was adjusted to maintain
a constant blockage ratio for all turbine geometries. Therefore, as aspect
ratio was increased by increasing turbine height $(H)$, the dynamic depth
$(h)$ was also increased. However this change in water depth decreased the
Froude number, which is given by

$Fr=\frac{U_{0}}{\sqrt{gh}},$ (3)

where $U_{0}$ is the inflow velocity, and $g$ is the acceleration due to
gravity. One interpretation of $Fr$ is the ratio of the flow’s kinetic energy
to its potential energy. To maintain $Fr$ while increasing the water depth,
$U_{0}$ was increased. However, increasing $U_{0}$ increased the Reynolds
number, defined with respect to the blade chord length as

$Re=\frac{U_{0}c}{\nu},$ (4)

where $c$ is the turbine blade chord length, and $\nu$ is the kinematic
viscosity of the working fluid. $Re$ represents the ratio of inertial forces
to viscous forces in a fluid flow. To hold $Re$ constant, the kinematic
viscosity was increased by decreasing the water temperature ($T$). In this
manner, we were able to hold all three parameters constant while varying only
the aspect ratio. While $Re$ can be defined in terms of other length or
velocity scales, this definition has the advantage of simplicity. For example,
incorporating turbine rotation into the velocity scale yields variations in
$Re$ with rotation rate and still neglects induction. As our focus is on rotor
hydrodynamics, the chord length was chosen as the length scale, but is
proportional to the diameter.

Experiments were conducted at a blockage ratio of 11.5% and at two $Re$-$Fr$
pairs: one "high pair" $(Re=4.27\times 10^{4},\,Fr=0.427)$ and one "low pair"
$(Re=2.03\times 10^{4},\,Fr=0.279)$. This was done to confirm that trends
observed at one combination of operating parameters were consistent with
another. The selected values of $Re$ and $Fr$ maximize the range of $Re$
achievable in the Tyler flume given the facility constraints on $h$, $U_{0}$,
and $T$, and the value for $\beta$ was chosen to be in the range of previous
experiments Strom (2019). Additionally, since controlling the Froude number in
cross-flow turbine performance experiments is not a universal practice, the
high $Re$-$Fr$ tests were repeated while holding only confinement and $Re$
constant to explore the significance of $Fr$ on results. The values of $h$,
$U_{0}$, and $T$ required to maintain $\beta$, $Fr$ (when held constant), and
$Re$ for each aspect ratio are given in Table 1. The turbulence intensity
($TI$) for each case is also reported. As turbine performance has been shown
to be sensitive to $TI$ primarily at higher levels (Maganga  _et al._ , 2010;
Blackmore, Myers, and Bahaj, 2016), we do not expect $TI$ to significantly
affect our results.

Table 1: Turbine geometries, flume conditions, and resulting non-dimensional
parameters for the three sets of experiments conducted.

Experiment | Turbine | Nominal Flume Parameters |  | Nominal Non-Dimensional
---
Flow Characteristics
| $AR$ |  $h$ (cm) |  $U_{0}$ (m/s) |  $T$ $(^{\circ}$C) |  $TI$111Defined as $\frac{\sqrt{2k/3}}{U_{0}}$, where $k$ is the turbulence kinetic energy of the inflow. (%) | $B\,(\%)$ | $Fr$ | $Re$
High $Re$-$Fr$ | 0.95 | 32.7 | 0.764 | 35.0 | 1.7 | 11.5 | 0.427 | $4.27\\!\times\\!10^{4}$
1.09 | 37.3 | 0.816 | 31.7 | 1.6
1.22 | 42.0 | 0.866 | 28.9 | 1.6
1.36 | 46.7 | 0.913 | 26.4 | 1.6
1.50 | 51.3 | 0.957 | 24.3 | 1.6
1.63 | 56.0 | 1.000 | 22.4 | 3.3
Low $Re$-$Fr$ | 0.95 | 32.7 | 0.500 | 20.2 | 2.2 | 11.5 | 0.279 | $2.03\\!\times\\!10^{4}$
1.09 | 37.3 | 0.535 | 17.5 | 2.2
1.22 | 42.0 | 0.567 | 15.2 | 1.9
1.36 | 46.7 | 0.598 | 13.3 | 1.7
1.50 | 51.3 | 0.627 | 11.5 | 1.6
1.63 | 56.0 | 0.655 | 10.0 | 1.4
High $Re$, variable $Fr$ | 0.95 | 32.7 | 1.000 | 22.4 | 1.5 | 11.5 | 0.559 | $4.27\\!\times\\!10^{4}$
1.09 | 37.3 | 1.5 | 0.523
1.22 | 42.0 | 1.6 | 0.493
1.36 | 46.7 | 1.7 | 0.467
1.50 | 51.3 | 1.8 | 0.446
1.63 | 56.0 | 3.3 | 0.427

###  II.3 Performance Metrics

The efficiency of the turbine, also known as the coefficient of performance or
power coefficient, is the ratio of power produced by turbine to the kinetic
power in the fluid that passes through the rotor swept area. The time-average
coefficient of performance is given by

$C_{P}\equiv\left\langle
C_{P}\right\rangle=\left\langle\frac{\tau(t)\omega(t)}{\frac{1}{2}\rho
AU_{0}(t)^{3}}\right\rangle,$ (5)

where $\tau(t)$ is the instantaneous hydrodynamic torque on the turbine,
$\omega(t)$ is the turbine’s instantaneous angular velocity, $U_{0}(t)$ is the
instantaneous freestream velocity, and $\rho$ is the density of the working
fluid. A Butterworth filter was applied to the torque measurements to
eliminate motor noise from the data. Because the ADV velocity measurement was
not synchronized with other data acquisition, an advection time correction was
applied to the inflow velocity (Polagye  _et al._ , 2019).

$C_{P}$ varies with the turbine’s tip-speed ratio, which is the ratio of the
turbine blade’s tangential velocity to the freestream velocity, given as

$\lambda=\frac{\omega R}{U_{0}},$ (6)

where $R$ is the radius of the turbine rotor. Turbines with each aspect ratio
were tested from $\lambda=0.8$ to $\lambda=3.2$ with a step size of 0.1. Data
was collected at each tip-speed ratio for 60 seconds, and the measurement time
series were trimmed to an integer number of turbine rotations.

Additionally, to isolate the performance of the turbine blades from the rest
of the rotor (i.e., the struts and central shaft), the parasitic support
structure torque was measured by repeating each experimental condition with a
bladeless turbine. The performance of the blades was then estimated as:

$C_{P,blade}\cong C_{P,turbine}-C_{P,supports}.$ (7)

This strategy, first proposed by Bachant  _et al._ (2016), assumes that
secondary interactions between the blades and support structures are limited,
which was borne out by Strom, Johnson, and Polagye (2018) for a range of
support structures, including the struts used here.

##  III Results

Figure 4: Percent error between target and measured time-average values for
the blockage ratio, Froude number, and Reynolds number shown for the high
$Re$-$Fr$ experiments (left) and the low $Re$-$Fr$ experiments (right). The
grey shading denotes $\pm 1.5\%$ deviation from the target value.

###  III.1 Parameter Control

Across experiments, $\beta$, $Fr$, and $Re$ were maintained within
approximately $\pm 1.5\%$ of target values, as shown in Fig. 4. The relatively
small variation in these parameters for a given aspect ratio can be attributed
to long-period ($>$ 10 s) oscillations corresponding to resonance between the
pumps and flume structure. The slight decrease in $\beta$, $Fr$, and $Re$ with
increasing $\lambda$ is due to greater rotor thrust at higher angular
velocities, which elevates the free-surface ahead of the rotor and reduces the
freestream velocity.

###  III.2 Turbine Efficiency

Figure 5: Time-average turbine efficiency versus tip-speed ratio across a
range of turbine aspect ratios for two $Re$-$Fr$ combinations.

As expected, turbine efficiency $(C_{P})$ depends on the $Re$-$Fr$
combination, but is insensitive to aspect ratio (Fig. 5). In other words, if
all other relevant non-dimensional parameters are held constant, the range of
aspect ratio has no effect on this turbine design. The increase in turbine
performance with increasing $Re$ at each aspect ratio is expected, as the
turbines were operated in a $Re$-dependent regime(Bachant and Wosnik, 2016b;
Miller  _et al._ , 2018).

The scatter in peak $C_{P}$ (defined as $\Delta C_{P}$) for the $Re$-$Fr$
combinations was 0.01 and 0.03 for the higher and lower combination,
respectively. As shown in Fig. 6a, this is further reduced when the
performance of the blade is isolated by subtracting the losses from the
support structures (Fig. 6b). We ascribe the residual scatter in the lower
$Re$-$Fr$ combination to low signal-to-noise ratio for torque.

Figure 6: Support structure losses (b) are subtracted from the turbine
efficiency (LABEL:{fig:cp3ParamFullTurb}) to estimate (a) blade-only $C_{P}$.
The grey curves in (a) show the full turbine performance from Fig. 5.

###  III.3 Froude Number Importance

Fig. 7 shows the percent deviation from target values of $\beta$ and $Re$ for
high $Re$ experiments repeated at varying $Fr$. As a consequence of varying
water depth, the Froude number increases by more than 30% from the largest to
smallest aspect ratio turbine.

Figure 7: Percent deviation between target and measured time-average values of
the blockage ratio, Froude number, and Reynolds number for the high $Re$ case
when $Fr$ is allowed to vary. For $\beta$ and $Re$, the grey shading denotes
$\pm 1.5\%$ deviation from the target values. Figure 8: Time-average $C_{P}$
versus tip-speed ratio across aspect ratio for (a) the full turbine and (b)
the turbine blades only for the high $Re$ and variable $Fr$. The grey curves
in (b) show the full turbine performance from (a).

Fig. 8a shows how $C_{P}$ increases substantially with decreasing aspect ratio
and increasing $Fr$. This trend is amplified for the blade-level performance
coefficient in Fig. 8b. As we have already demonstrated that performance is
independent of aspect ratio when $Fr$ is held constant, the observed change is
solely a consequence of increasing $Fr$. The increased efficiency with $Fr$ is
consistent with simulations of cross-flow turbines by Consul, Willden, and
McIntosh (2013) and Gauvin-Tremblay and Dumas (2020).

As a consequence of holding the blockage ratio constant for these experiments,
when the aspect ratio was decreased, the dimensional distance between the
turbine and the free-surface was also decreased. For experiments with variable
$Fr$, we observed increasing ventilation of the downstream turbine blade (see
Fig. 9). Although the torque contributed by the downstream blade is small,
ventilation would be expected to increase the form drag on the blade (Young
_et al._ , 2017) and, consequently, reduce the net rotor torque. Therefore,
the performance increase with $Fr$ may have been even larger if not for
ventilation losses.

Figure 9: The most severe ventilation observed on the downstream sweep of the
blade ($AR=0.95$ for the variable $Fr$ case).

##  IV Discussion

###  IV.1 Effect of Aspect Ratio on Performance

We find that, when other non-dimensional parameters are held constant, aspect
ratio has no effect on $C_{P}$ in the range of $Re$, $Fr$, $\beta$, and $AR$
tested. This outcome is consistent with the underlying fluid mechanics. The
only mechanism for aspect ratio to affect turbine performance is through the
relative extent of the blade span affected by tip losses or support effects.
In our experiments, strut and tip losses were relatively minor compared to
power production by the blades (Fig. 6). With further reductions in $AR$, we
would eventually expect to see $C_{P}$ decline as those losses become
appreciable.

Additionally, with a different blade support architecture, we could anticipate
more significant effects at higher aspect ratio. For example, mid-span struts
have significant, undesirable blade-strut interactions (Strom, Johnson, and
Polagye, 2018), shown by Villeneuve and Dumas (2019) to be a consequence of
inopportune vortex shedding at the blade-strut interface. Li  _et al._ (2017)
employed mid-span struts and, consequently, found performance to increase with
aspect ratio, since as the blade span increased, relatively less of the blade
was impacted by this phenomenon. Our conclusion that changes in turbine
performance with aspect ratio are primarily a consequence of strut
interactions is, therefore, consistent with Li et al., who were also rigorous
in holding other non-dimensional parameters constant in their simulations.
Consequently, we would expect the efficiency of turbines with blade-end
supports that produce higher parasitic torque, such as the end-plates and
thicker struts (Strom, Johnson, and Polagye, 2018), to increase with aspect
ratio.

###  IV.2 Assessment of Experimental Approach

The support structure interactions responsible for changes in performance with
aspect ratio are inherently three-dimensional. Therefore, an analytical or
numerical study exploring this parameter space must either be three-
dimensional or include empirical models for tip and support losses to extend a
two-dimensional study. Because of this, the experimental approach taken here
is a cost-effective exploratory tool. However, in experiments, unlike in
simulation, it can be challenging to hold all other relevant non-dimensional
parameters constant. By controlling $\beta$, $Fr$, and $Re$, we observed non-
dimensional performance similarity (i.e., consistent $C_{P}$-$\lambda$
dependence) across the range of aspect ratios tested. While this approach
limited the range of $AR$ that could be explored, failing to do so would have
led to erroneous conclusions about the effects of $AR$. As demonstrated, if
the Froude number was allowed to vary, we could have incorrectly attributed an
increase in turbine efficiency to a decrease in aspect ratio. This would have
been a baffling result defying physical intuition. More broadly, this
demonstrates that the Froude number is an important consideration for cross-
flow turbine experiments conducted in an open channel. We note that there
remains some ambiguity in the appropriate definition of $Fr$, as $Fr$ defined
in terms of the separation between the top of the rotor and free surface also
remained constant during our experiments.

Finally, rotor ventilation, particularly of the downstream sweep, was observed
in some experiments, but not explicitly accounted for in any of the non-
dimensional parameters we held constant. As ventilation was most severe for
combinations of relatively high flow speeds and low water depth, this is
likely to be an important consideration at relatively high Froude numbers. The
non-dimensional similarity in turbine performance when $\beta$, $Fr$, and $Re$
were held constant demonstrates that ventilation, even when observable, does
not necessarily have a meaningful effect on turbine performance. Further
investigation of rotor ventilation on cross-flow turbine performance may be
appropriate.

##  V Conclusion

Optimizing cross-flow turbine geometry could increase their cost-
effectiveness. However, to draw firm conclusions about the effects of a
specific geometric variable, all other non-dimensional parameters that impact
turbine performance must be held constant. Due to a lack of generalized
analytical models and the computational expense of three-dimensional numerical
simulations, experimental exploration is an attractive approach for exploring
geometric variability. Here, as an example, we experimentally assess the
effect of rotor aspect ratio on turbine performance while rigorously
maintaining the confinement, Froude number, Reynolds number, and other non-
dimensional rotor parameters (e.g., chord-to-radius ratio).

For the straight-bladed cross-flow turbine considered here, performance is
shown to be independent of aspect ratio from 0.95 to 1.63 when other non-
dimensional parameters are held constant. The mechanism for changes in turbine
performance with aspect ratio is the balance between losses due to blade
supports or blade tip vortices and power production by the blades. Our choice
of supports (blade-end struts with a thin, foil cross-section) contributed to
the aspect ratio invariance for the turbine. When support structure losses are
accounted for, blade-level performance is almost entirely independent of
aspect ratio. Aspect ratio is, therefore, most likely to affect turbine
performance when support structures incur greater losses, such as with end-
plates, or produce undesirable blade-strut interactions, such as with mid-span
supports. At even lower aspect ratios than those studied here, we would expect
support structures to have more pronounced effects since, as the aspect ratio
approaches zero, effects on blade-level performance through blade-strut
interactions would no longer be negligible.

This study demonstrates the importance of isolating the effects of non-
dimensional parameters from the geometric parameter under investigation. In
our experiments, we held confinement, Froude number, and Reynolds number
constant across all aspect ratios. However, decreasing aspect ratio while
holding only confinement and Reynolds number constant results in an increasing
Froude number and an associated increase in turbine efficiency. The magnitude
of this effect, a turbine efficiency increase of 12% for a Froude number
increase of 30%, highlights the importance of controlling for this parameter
in experiments.

## Acknowledgements

This work was supported by the United States Department of Defense Naval
Facilities Engineering Command (NAVFAC) under contract N0002418F8702. The
authors would like to thank Paul Gibbs for prompting the inception of this
study, Eamon McQuaide and Reginald Rocamora for their assistance in
manufacturing the turbine blades used in these experiments, and Erik Skeel for
machining the struts. A special thanks is extended to Benjamin Strom and
Hannah Ross for designing and continually improving the cross-flow turbine
experimental set-up that was used in this investigation.

## Data Availability

Turbine performance data supporting this study, as well as a video of the
observed rotor ventilation, can be accessed at ResearchWorks, the University
of Washington’s digital repository. This material may be accessed at
http://hdl.handle.net/1773/45618.

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