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RESEARCH ARTICLE
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MARCH 17 2022
Parametric study of small-scale rotors in axial descent
Marcel V
eismann
;
Daniel Y
os
;
Morteza Gharib
Physics of Fluids
34, 035124 (2022)
https://doi.org/10.1063/5.0083761
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Parametric study of small-scale rotors in axial
descent
Cite as: Phys. Fluids
34
, 035124 (2022);
doi: 10.1063/5.0083761
Submitted: 29 December 2021
.
Accepted: 25 February 2022
.
Published Online: 17 March 2022
Marcel
Veismann,
1,a)
Daniel
Yos,
2
and Morteza
Gharib
1
AFFILIATIONS
1
Graduate Aerospace Laboratories, California Institute of Technology, 1200 E. California Blvd., Pasadena, California 91125, USA
2
California State University Long Beach, 1250 Bellflower Blvd., Long Beach, California 90840, USA
a)
Author to whom correspondence should be addressed:
mveisman@caltech.edu
ABSTRACT
Despite extensive research in multirotor aerodynamics in the recent past, axial descent, specifically the vortex ring state, still poses great
challenges for multirotor configurations as this flight stage is typically accompanied by severe losses in rotor thrust and strong thrust
fluctuations. This paper presents a parametric study to investigate the influence of relevant geometric parameters of a small-scale rotor blade
on the rotor performance in axial descent. Design variables subject to variation were the collective pitch, chord length, taper ratio, number of
blades, as well as the tip geometry. Custom rotors for each parameter modification were manufactured and experimentally evaluated in wind
tunnel tests with mean thrust recordings and measurements of the thrust fluctuations serving as performance metrics. Results indicated that
rotor blades with larger aspect ratio and higher blade loading coefficient are less affected by the adverse aerodynamics in the vortex ring state,
experiencing lower thrust losses and vibrational loads. Particle image velocimetry flow visualization confirmed that the aerodynamic losses in
the vortex ring state can be attributed to blade vortex interactions. Comparison of the rotor flow structure in hover of all investigated rotor
designs suggested that improvements in the descent performance of a rotor stem from a combination of reduced tip vortex strength and
increased axial tip vortex convection rate. Using the experimental findings of this study, a predictive model for approximating the maximum
extent of mean thrust losses in axial descent for a given blade geometry and hover thrust coefficient could be established.
Published under an exclusive license by AIP Publishing.
https://doi.org/10.1063/5.0083761
I. INTRODUCTION
Multirotor craft, easily adaptable and low-cost vehicles, have
established themselves as valuable platforms for academia, industry,
and consumers in the recent past. However, despite extensive research
to improve flight characteristics and optimize performance, the chal-
lenges regarding axial descent of small-scale rotors have to date
remained largely unaddressed. That is, while descending at a steep
angle, the rotor flow downwash is re-ingested through the actuator
disk, which greatly compromises the thrust generation of a rotorcraft.
At descent rates close to the rotor induced velocity, the recirculation of
the rotor flow results in the formation of a toroidal vortex ring system
around the rotor disk, leading to the most pronounced performance
losses across the axial descent regime. This particularly critical flow
condition is commonly referred to as the
vortex ring state
(VRS).
1,2
Based on available research on small-scale rotors, significant reduc-
tions in mean rotor thrust for a given rotation rate can be expected
when operating under VRS conditions.
3
Additionally, a strong low-
frequency oscillatory behavior of the airloads is generally observed,
4,5
leading to severe vibrations that can ultimately manifest themselves in
large vehicle attitude oscillations.
6
This performance degradation and
the increased vibrational loads in axial descent can considerably limit
the controllability and operational margins of multirotor platforms.
While prior research has extensively investigated the nature of
the unsteadiness in the disk loading of rotorcraft in the VRS, it was
done so predominantly with respect to large-scale helicopter rotors.
7
Accordingly, the fundamental fluid mechanics of the VRS are mostly
understood, and it is generally believed that blade-vortex interactions
(BVIs) are the leading cause for rotor performance losses. (A more
detailed explanation is provided in Sec.
II
.) Meanwhile, this subject
matter is recently gaining renewed interest with a shifted focus toward
low Reynolds number rotors as the VRS phenomenon has shown to
pose similar challenges for the emerging multirotor systems as it does
for large-scale helicopters.
6
To date, large amounts of research works have been dedicated
toward blade geometry optimizations of small-scale rotors for hover
and forward flight conditions,
8–12
but optimizations for axial descent
have so far been given little consideration. A focus on hover and for-
ward scenarios in the rotor blade design process is understandable as
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
34
, 035124-1
Published under an exclusive license by AIP Publishing
Physics of Fluids
ARTICLE
scitation.org/journal/phf
06 October 2023 18:43:59
these flight stages typically constitute the majority of a multirotor’s
flight profile. The descent stage, on the other hand, generally assumes
a much smaller part in comparison, and efficiency in this flight stage
is, therefore, less critical for the system’s overall range/endurance.
However, the incentive for better adapting a rotor for descent scenar-
ios is likely not driven by efficiency considerations but much more by
safety concerns, since descent is typically considered the most hazard-
ous flight stage. So far, without adequately adapted rotors for descent,
operational precautionary measures have predominantly been used,
such as limiting the descent rates of a multirotor vehicle, to ensure safe
descent operation. While these operational measures present a viable
alternative as they circumvent the complications of entering into the
VRS at higher descent rates altogether, they can become restrictive
and may fail under unforeseen circumstances such as wind gusts.
Furthermore, when solely focusing the rotor design process on hover
conditions and disregarding any descent considerations, future rotor
designs may be driven further toward unstable descent performance,
where even operational measures cease to be effective. Thus, an in-
depth analysis on how the overall rotor design, comprising a multitude
of geometric parameters, determines the descent characteristic is cer-
tainly invaluable. This knowledge will provide a useful framework for
designing rotors that are inherently more stable and enable more
aggressive descent rates. Furthermore, findings can also help to guide
these operational guidelines (i.e., predicting the maximum allowable
descent rates) for a given rotor design. Consequently, a growing stream
of research is slowly being dedicated toward more comprehensive
characterization of fluid mechanics around small-scale rotors in the
VRS and the implications on the thrust generation. However, existing
studies of small-scale rotorcraft operating in the VRS are largely fun-
damental and investigative in nature, providing insights into the fluid
dynamics of the specific rotor blade geometries used within the study
without systematically identifying the significance of design parame-
ters of these fixed-pitch rotors and with no attempts to improve or
optimize the descent performance.
4
Meanwhile, only few studies to
date were specifically targeted at investigating the influence of selected
design parameters on small-scale propellers such as the pitch.
3,13
Hence, until now, not enough data are available to reliably establish
the influence of the specific blade variables on VRS behavior,
14
espe-
cially in regard to low Reynolds number rotors commonly employed
on multirotor systems.
Given this lack of information, this paper presents a parametric
analysis to provide an experimental assessment of the influence of rele-
vant rotor blade design parameters on the descent performance of a
small-scale rotor. The sensitivity analysis was conducted by indepen-
dently varying the collective pitch, taper ratio, chord length, number
of blades as well as the tip geometry of a rectangular reference blade
with linear twist. A separate rotor for each parameter variation was 3D
printed and evaluated under simulated descent conditions in wind
tunnel tests. Measurements of the generated mean thrust and thrust
fluctuations at constant rotation rate were used to characterize the
rotor performance. In total, 18 different rotor geometries were investi-
gated. Particle image velocimetry (PIV) flow visualization was per-
formed as well to provide further insights into the flow characteristics
around small-scale rotors in axial descent and to quantify differences
in the rotor flow structure of the various rotor designs. While previous
studies have started to assess the extent to which selected geometric
parameters determine the vertical descent characteristics of a rotor,
this study explores a so far unmatched parameter space. By utilizing
custom, 3D printed rotors, where each geometric variable could be
precisely controlled and modified completely decoupled from the
remaining geometry, the results enable a direct and extensive compari-
son between all investigated parameters. The inclusion of concomitant
flow visualization data further aids in comparing different rotor
designs and provides an explanation for the observed trends in the
descent characteristics. While this study primarily aims at comprehen-
sively establishing the dependence of rotor descent characteristics
upon its geometric variables, it also seeks to expand the fundamental
understanding of the fluid mechanics causing the unsteadiness in the
VRS. Findings of this study may find practical applications in the
design process of future rotor blades by providing guidelines for more
stable multirotor platforms. Moreover, these results can have addi-
tional implications for rotorcraft noise reductions, as the dominating
aerodynamic rotor noise is similarly caused by BVI.
15
II. ROTOR AERODYNAMICS IN VERTICAL FLIGHT
The axial flight regime of a rotor can be classified into four funda-
mental flow states based on the definitions of the momentum theory,
which are illustrated in
Fig. 1
.Inthe
normal working state
,corre-
sponding to ascent and hover (
v
c
0), the induced fluid velocity at
the rotor plane and power are given by the momentum theory
1
v
i
¼
1
2
v
c
þ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
2
c
þ
2
T
q
A
s
;
(1a)
P
¼
Tv
¼
T
ð
v
c
þ
v
i
Þ¼
1
2
Tv
c
þ
T
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
2
c
þ
2
T
q
A
s
;
(1b)
where
v
i
is the rotor induced velocity,
v
c
is the climb velocity,
T
is the
rotor thrust, and
A
is the disk area (i.e., the area swept out by the rotor
blades). Under hover conditions (
v
c
¼
0), Eq.
(1a)
yields
v
h
¼
ffiffiffiffiffiffiffiffiffi
T
2
q
A
s
;
(2)
which is the induced velocity of a rotor in hover and often serves as a
velocity reference. In axial descent (
v
c
<
0), where the relative free-
stream velocity is directed upward, a different descent model of the
momentum theory needs to be applied to calculate the induced
velocity
v
i
¼
1
2
v
c
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
2
c
2
T
q
A
s
:
(3)
This descent model, however, only provides physical values for high
descent rates of
v
c
<
2
v
h
(
windmill brake state
), where all flow is
nominally directed upward and a distinctive rotor flow slipstream
exists. For moderate descent rates of
2
v
h
<
v
c
<
0, the analytic
expressions of the momentum theory no longer hold, because the flow
direction within the rotor slipstream is not clearly defined and no
definitive control volume can be established.
2
The two flow states
in this critical descent rate range are distinguished based on their
nominal flow direction through the disk. At lower descent rates, with
v
i
>
j
v
c
j
and flow nominally downward through the rotor, the flow
pattern around the disk resembles concentric sets of vortex rings,
Physics of Fluids
ARTICLE
scitation.org/journal/phf
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
34
, 035124-2
Published under an exclusive license by AIP Publishing
06 October 2023 18:43:59
giving this state the name vortex ring state (VRS). In the VRS, the flow
at the rotor exhibits large amount of recirculation and high turbulence
leading to average thrust losses and strong thrust fluctuations. As
descent rates increase and the flow direction through the rotor changes
the sign to being upward, the unsteadiness of the flow pattern shifts to
the wake above the rotor, known as
turbulent wake state
(TWS). The
TWS is still a highly unsteady flight regime; however, thrust losses and
vibrations are considerably lower than in the VRS. Therefore, the VRS
is generally considered the most critical operating stage.
Based on previous studies, it is generally believed that the oppos-
ing freestream in axial descent reduces the vorticity transport of the
rotor flow leading to conditions, where tip vortices remain and accu-
mulate in the rotor plane
16
and introduce strong BVIs.
2
This is because
when the trailed vorticity of a rotor blade remains in close proximity
to the rotor, it can induce relatively high velocities with upwash and
downwash components, disturbing the inflow conditions of the fol-
lowing blade. These vortex-induced upwash and downwash velocities
have been shown to be quite significant for small-scale rotors and can
be as much as 25% of the rotor tip speed. The relative tip vortex
strength on small-scale rotors has also been found to be much larger
than those found on larger scale rotors,
17
suggesting that rotor blades
of this smaller size may be particularly affected by VRS aerodynamics.
The VRS-flow-field is, furthermore, marked by a periodic collapse of
the rotor wake, when the vorticity accumulation in the rotor plane
becomes too large. The collapse of the rotor wake then allows the
trapped tip vortices to escape, clearing the vorticity build-up from the
rotor plane. Subsequently, as the rotor wake reestablishes, the accumu-
lation starts again. This repetitive process of vorticity build-up and
shedding leads to the distinctive low-frequency thrust fluctuations.
1
Experimental studies on small-scale rotors have indicated that the
oscillatory behavior of the airloads occurs at frequencies that are on
the order of multiple rotor rotations.
4
Given that the axial component
of the vortex trajectories is predominantly determined by the magni-
tude of the rotor induced velocity and the opposing freestream, the
non-dimensional descent rate ratio
v
c
=
v
h
is considered the primary
scaling for VRS aerodynamics.
14
However, it is clear that specific geo-
metric and operational parameters of the rotor blade can have a sec-
ondary, albeit not insignificant, impact: twist, planform, number of
blades, solidity, collective, tip speed, Reynolds number, and Mach
number. Although there have been multiple studies recently regarding
the influence and significance of isolated blade parameters such as
pitch, twist, and solidity (e.g., Refs.
13
and
18
)resultsareeithersparse
or inconclusive, especially in regard to small-scale, low Reynolds num-
ber propellers.
Not limited to axial flight, further relevant non-dimensional
quantities used with this study for rotor comparisons are the thrust
coefficient and power coefficient, which are defined as
C
T
¼
T
qp
R
2
ð
X
R
Þ
2
¼
T
q
A
ð
X
R
Þ
2
;
(4)
C
P
¼
P
qp
R
2
ð
X
R
Þ
3
¼
P
q
A
ð
X
R
Þ
3
:
(5)
An efficient strategy for establishing these coefficients for small-scale
rotors is to use dedicated thrust stands and least-square fitting mea-
surements of generated thrust,
T
,andpower,
P
, as a function of rota-
tion rate
X
. The rotor efficiency is then expressed by the
figure of
merit
, which is the ratio of ideal power of hovering to the actual
mechanical power to drive the rotor and is defined as
2
FM
¼
P
ideal
P
actual
¼
C
3
=
2
T
ffiffiffi
2
p
C
P
:
(6)
The thrust coefficient can, furthermore, be reduced by the rotor’s
solidity,
r
, resulting in the blade loading coefficient
C
T
=
r
for a better
description of the local lift loading
2
with the rotor solidity being
r
¼
A
b
A
N
b
cR
p
R
2
;
(7)
where
A
b
N
b
Rc
denotes the total bladed area of a rotor. In this study,
the bladed area was chosen based on the rotor’s planform area, excluding
the rotor hub. It should be noted that the definition of
r
can be quite sen-
sitive to the selection of the appropriate blade area for these rotors with rel-
atively large rotor hubs. The rotor aspect ratio can then be defined as
AR
¼
R
c
N
b
R
2
A
b
:
(8)
FIG. 1.
Schematic flow fields of a rotor in different working states in axial flight (adapted from Ref.
2
) normal working state (left), vortex ring state (center-left), turbulent wake
state (center-right), and windmill brake state (right).
Physics of Fluids
ARTICLE
scitation.org/journal/phf
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
34
, 035124-3
Published under an exclusive license by AIP Publishing
06 October 2023 18:43:59
III. INVESTIGATED ROTOR BLADE DESIGNS
Based on the presumption that the adverse rotor aerodynamics
related to the VRS are a consequence of vortex induced inflow distur-
bances, it is believed that reducing the tip vortex strength or increasing
vorticity transport away from the actuator disk may have a beneficial
impact on the rotor performance in axial flight. The design of small-
scale rotor blades should, therefore, have intrinsic potential for
improving the descent performance by seeking means to attenuate the
BVI. Since loading on a rotor blade is highly concentrated in the tip
region, the design of the rotor tip geometry lends itself as one critical
design parameter for performance improvements.
19
In practice,
advanced blade tip shapes generally attempt to modify the tip vortex
structure for performance enhancement and/or noise/vibration reduc-
tion.
20
In this context, we investigated how various tip shapes can help
to moderate tip vortex strength or alter evolutional characteristics for
minimizing vorticity deposition in the rotor plane during the VRS to,
thus, reduce performance losses and roughness in this flight stage.
Beyond examination of selected tip shape modifications, a parameter
space of relevant blade geometry variables and their significance on
the descent performance was explored, which has up-to-date only
sparsely been attempted, particularly for low Reynolds number rotor
operation.
Figure 2
schematically illustrates selected rotor designs investi-
gated throughout the study. All rotors were manufactured using a
Formlabs Form 3
stereolithographic 3D printer and the
Formlabs
Rigid 4000
resin
21
at a print layer height of 100
l
m. This glass-filled
resin combines a smooth, polished surface finish with high stiffness
and is well suited for aerodynamic parts that can withstand high loads
at minimal deflection. Selected manufacturer-specified material prop-
erties are listed in
Table I
. While the printed parts have a smooth sur-
face finish, exact surface roughness values could not be determined.
To verify that the rotor manufacturing process itself does not intro-
duce noticeable aerodynamic effects, resin material and surface rough-
ness (via export settings for the printer file) were varied for the
reference rotor with results only showing minimal changes on the
overall rotor performance in descent (see the
supplementary material
).
The reference rotor, serving as the baseline, was a two-bladed
rotor with rectangular blades, a diameter of 6 in. (152.4mm), linear
twist from root to tip, and had a NACA0012 airfoil. A rotor size of
6 in. was chosen as it was the largest printable size within the 3D
printer dimensions. Relevant geometric parameters of this reference
blade are listed in
Table II
. All other blade designs were modifications
of this baseline with each variant exploring a single, isolated parameter
variation (or tip change). For consistency, all investigated rotors were
designed with the same diameter of 6 in. (152.4 mm)—keeping the
disk area constant—and used the same NACA0012 airfoil. An over-
view and more detailed description of the various investigated rotor
blade designs are provided in
Table III
. In this context, it is convenient
to group the rotor blade modifications into three categories: (1) blade
tip changes, including sweep, anhedral and dihedral winglets, Ogee,
and Vane tips,
20
(2) variations of relevant geometric parameters
including chord length, taper ratio, and collective pitch, and (3) rotors
with different blade counts, namely, a four bladed rotor. A commercial
blade (
Com
) was also tested to examine how off-the-shelf rotor blades
compare to these 3D printed ones. This rotor had the designation
6
3 (diameter x pitch) and was provided by
HQProp
.Sincetheexact
rotor geometry was not fully known, its results are not discussed in
greater detail but are merely used for comparison purposes. To help
visualize the geometry of the various rotors with alternative blade tip
design in greater detail and highlight differences between tip modifica-
tions, isometric views of the computer aided design (CAD) models
used to print the rotors
b1
–
b5
are compared to the reference rotor
(
Ref
.) in
Fig. 3
.
The selection of geometric parameters to modify was primarily
guided by previous studies on tip vortex characteristics of rotating
wings
2,3,6–8,10,13,14,19,20,22–26
with the explored blade parameters (and
tip shapes) being the most promising solutions for improving VRS
behavior by reducing the extent of BVIs. A brief review of each blade
parameter (and tip shape) is given in the following, placing a particular
emphasis on the influence on tip vortex characteristics and figure of
merit (i.e., hover efficiency):
•
Tip sweep
: Rotor blade tip sweep has been introduced as a popu-
lar design to delay compressibility drag rise and, therefore, reduce
FIG. 2.
Selected top and side views of investigated rotor blade designs.
TABLE I.
3D printer resin material properties.
21
Property
Value
Ultimate tensile strength
69 MPa
Tensile modulus
4.1 GPa
Elongation at break
5.3%
Flexural strength
105 MPa
Flexural modulus
3.4 GPa
TABLE II.
Key parameters of the reference rotor.
Parameter
Value
Airfoil
NACA 0012
Blade count
2
Chord length
15 mm
Rotor diameter
6 in. (152.4 mm)
Root pitch
26
Tip pitch
11
Re
tip
;
max
(at 10 000 RPM)
7
:
90
10
4
Physics of Fluids
ARTICLE
scitation.org/journal/phf
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
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, 035124-4
Published under an exclusive license by AIP Publishing
06 October 2023 18:43:59
the rotor power requirements. A proven parametrization is an
approximately 20% sweepback at 0.85R.
19
Previous computa-
tional optimization studies showed that sweep decreases profile
power requirements due to lower effective tip Mach numbers,
and the reduced effective chord of the blade tip can result in a
more uniform lift distribution.
8
Meanwhile, experimental mea-
surements on sub-scale models showed that the lift distribution
and vortex strength are close to that of a rectangular blade, but
vortex cores appeared to be trailed further outboard and away
from the rotor.
22
Hence, blade tip sweep is believed to alter tip
vortex formation and overall vortex structure.
2
It should be noted
that tip sweep is predominantly employed for full-scale rotor
blades, where tip Mach numbers are significant.
•
Winglet (dihedral/anhedral)
: Blade tip winglets on wind turbine
blades have shown to be a viable solution for reducing the effect
of vorticity trailing on rotating wings by reduction in spanwise
flow at the tip.
23
On full-scale helicopters, however, large angles
of dihedral have so far shown to be problematic, particularly in
forward flight.
20
Conversely, small amounts of anhedral have
been shown to provide minor hover efficiency improvements due
to reduced loading in the tip region and a larger downwards dis-
placement of tip vortices in the wake. Consequently, anhedral
tips are sometimes employed on rotor blades in complex, often
swept and tapered, geometries. Numerical studies confirmed that
small angles of anhedral can beneficially impact hover perfor-
mance while being less efficient in forward flight.
24
Unlike their
large-scale counterparts, a study on low-Reynolds number micro
air vehicles found that employing small winglets at heights of
about 6% of the rotor radius promotes minor improvements in
hover efficiency.
25
It was concluded that this can be attributed to
the winglet diffusing the tip vortices, and improvements from
upwards and downwards winglets were reported to be compara-
ble. Experiments of this study showed that the sizing of winglets is
determined by competing factors of induced power reduction, via
more effective tip vortex diffusion, and increasing profile power.
•
Ogee
: The Ogee tip was designed as an option for low-noise rotor
blades due to reduction of peak swirl velocities of the tip vortex,
i.e., creating a more diffused tip vortex. In terms of its effects on
performance, it is unclear whether the reduction of area at the tip
is advantageous for improving the figure of merit or if this design
promotes early stall and results in growth in profile power.
20
•
Vane tip
: Vane tips follow the approach of splitting the tip vortex
into two separate, but smaller vortices and are traditionally used
to reduce BVI-induced rotor noise. Smoke visualization experi-
ments have shown that these twin vortices can remain distinct
for a relatively long time without merging.
20
It is so far unclear
whether or not generating smaller, but more tip vortices is
advantageous for the axial descent behavior. However, the effi-
ciency is likely negatively impacted by this tip modifications, as
the area near the tip is reduced significantly along with the tip
chord Reynolds number. Given that the maximum rotor radius
was specified to the same value of 6 in. (152.4 mm) for all rotors,
introducing the tip vane increases wetted area in comparison to
the effective rotor area, which yields higher skin friction drag for
a given thrust.
•
Chord length
: According to Eq.
(7)
, rotor solidity can be regulated
via chord length or the number of blades. Solidity on full-scale
helicopters generally varies from about 0.08 to 0.12 (Ref.
2
), where
decreasing solidity usually leads to
FM
improvements, but reduces
stall margins. For small-scale, low Re rotors, on the other hand,
experimental investigations showed that higher solidity by increas-
ing the chord length (from
r
¼
0
:
17 to considerable
r
¼
0
:
32)
had a beneficial effect on the figure of merit.
10
While solidity
changes through chord length modifications showed significant
FM
improvements, changes by increasing the number of blades
resulted merely in minor performance gains. The observed perfor-
mance gains due to elongated chord lengths are believed to derive
from increased Reynolds number or more uniform inflow distri-
bution arising from 3D-effects from smaller aspect ratios.
25
The
reason for these contradicting findings between full-scale and
small-scale rotors is that the performance for small-scale rotors
with short chords and tip Reynolds numbers under 10
6
is particu-
larly sensitive to changes in the Reynolds number.
2
While increas-
ing chord length is anticipated to yield an overall higher figure of
merit by increasing the Reynolds number, the reduced aspect ratio
promotes a greater induced drag and tip vortex strength.
•
Taper
: Blade taper, generally employed in combination with
blade twist, is used for designing the optimum rotor with mini-
mal induced losses.
20
The primary effect of a tapered blade
TABLE III.
Rotor blade parameter variation.
Identifier
Variation
Description
b1
Sweep
20
sweep-back at 0.85R
b2
Anhedral
Blade tip anhedral at 0.85R
b3
Dihedral
Blade tip dihedral
b4
Ogee
Ogee tip at 0.85R
b5
Vane
Vane tip at 0.85R
b6.1–b6.2
Chord length
Chord length variation in mm
(
2,
þ
5)
b7.1–b7.3
Taper
Linear blade taper, taper ratio
(1.2, 1.5, 3)
b8.1–b8.6
Pitch
Collective pitch variation in deg.
(
6,
4,
2,
þ
2,
þ
4,
þ
6)
b9
Number of blades Four bladed rotor, identical blade
geometry as
Ref
.
FIG. 3.
Isometric views of rotor tip modifications.
Physics of Fluids
ARTICLE
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, 035124 (2022); doi: 10.1063/5.0083761
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planform is to achieve a more uniform lift distribution by
decreasing bound circulation near the blade tip and by loading
the inboard region more heavily. In theory, this results in a more
uniform inflow leading to the reduced tip vortex strength for a
given blade loading
22
and improvements in the figure of merit.
To study the effect of blade tapering on VRS behavior, we opted
for maintaining a constant root chord and employing a simple
linear taper by decreasing the tip chord length. This approach
required the least modifications of the reference blade. The taper
ratio was defined as the ratio of the two chord lengths
TR
¼
c
root
c
tip
:
(9)
An alternative approach would be to maintain a constant thrust
weighted blade solidity, which is defined as
8
ð
R
0
r
2
c
ð
r
Þ
dr
¼
const
:
(10)
This approach, however, would require a more extensive modifi-
cation of the reference blade geometry than just simply reducing
the tip chord length and could potentially introduce other effects
as well.
•
Blade pitch
: During experimental full-scale flight tests, increasing
blade pitch (i.e., collective pitch on full-scale helicopters) has
been reported to mitigate, and at times even suppress, VRS
behavior.
14
Studies on small-scale multirotor platforms corrobo-
rated these findings by showing that increased collective pitch
can reduce variation in the thrust for a given rotation rate in ver-
tical flight.
3,6,13
•
Number of blades
: The sparse amount of available data does not
support drawing any reliable conclusions about the influence of
the number of blades on the VRS behavior of a rotor.
14
In this
regard, the effect of increasing the blade count may have conflict-
ing aerodynamic characteristics: larger number of blades leads to
weaker tip vortices for a constant rotor thrust, thus potentially
reducing the magnitude of disturbances caused by interaction of
tip vortices and the rotor blade in the VRS. However, with more
blades, which are closer azimuthally, the number of BVI events
also increases, and blades are more likely to interact with vortices
of the previous blade.
2
Meanwhile, hover efficiency of a rotor has
shown to be relatively insensitive to the number of blades given
certain constraints based the Reynolds number it is operating at:
for low Reynolds number rotors if the constant chord length
remains constant (and thus varying disk solidity)
25
and for full-
scale systems, if solidity remains constant (by decreasing chord
length).
2
Consequently, the number of blades in helicopter
designs is primarily determined by dynamic considerations,
rather than aerodynamics.
•
Rotor thrust
: Previous experiments noted that variations in rotor
thrust had no significant impact on the inflow properties of
rotors in the VRS.
7
This is strictly speaking, however, only true if
thrust variation occurs by adjustment of rotation rate (without
drastically changing the Reynolds number), thus keeping rotor
parameterization fixed as explained in the previous item.
Meanwhile, thrust modulation at constant rotation rate and,
therefore, by changing the thrust coefficient via collective pitch
has shown potential influence on VRS behavior, where rotors
with lower pitch (and lower thrust coefficient) showed larger var-
iations in the inflow curves during axial descent.
26
IV. METHODOLOGY
A. Experimental apparatus
The aerodynamic descent performance of the examined small-
scale rotors was evaluated in wind tunnel experiments. The experi-
mental setup and its key parameters are schematically illustrated in
Fig. 4
. The rotors were placed in a vertical multi-fan wind tunnel facil-
ity to simulate the relative freestream velocity of a steady, axial descent.
This open-jet wind tunnel is composed of 121 DC fan units and
can deliver flow speeds of up to 9.6 m/s. The test section size is
1054
1054 mm
2
and much larger than the rotor dimensions, which
is why wind tunnel edge effects should be insignificant. While each fan
of this wind tunnel can be individually controlled, all fans were
assigned identical duty cycles in this study for a uniform flow field. A
flow straightener, perforated plates, and wire meshes were installed for
flow conditioning, and turbulence intensity values of approximately
0.4% were measured in the part of the flow, where the rotor assembly
was located. Velocity and turbulence intensity measurements were
performed prior to test runs and in the rotor plane with no rotor pre-
sent to correlate the freestream velocities to the wind tunnel duty cycle.
For consistency with published rotorcraft literature presented in Sec.
II
, the naming convention for velocity directions was that positive
velocities were directed downward (i.e., the rotor flow,
v
i
, is positive,
while the vertically rising flow of the wind tunnel,
v
c
, is negative).
Rotor thrust measurements were obtained by a one-dimensional
load cell at 2000 Hz, which was located upstream of the rotor assembly
and a support structure provided the connection to the rotor assembly.
Minimal flow interference by the support structure assured that results
were not masked by fluid-structure interactions and allowed to conclu-
sively establish the rotor performance as a function of the descent rate.
Therefore, designs were chosen without components in the rotor
wake and wind tunnel inflow. The rotors also operated in the pusher
configuration to further minimize the obstruction of rotor wake and
minimize any potential structure related aerodynamic influence.
FIG. 4.
Experimental setup of a rotor in the vertical multi-fan wind tunnel (flow
manipulators and structural elements not displayed for illustration purposes).
Physics of Fluids
ARTICLE
scitation.org/journal/phf
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, 035124 (2022); doi: 10.1063/5.0083761
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, 035124-6
Published under an exclusive license by AIP Publishing
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This design was anticipated to closely resemble idealized rotors in axial
flight, which are often modeled without any auxiliary structure and
with an undisturbed freestream flow. The investigated rotors of inter-
est were placed sufficiently downstream in the test section (approxi-
mately 1m off the wind tunnel fans) face to circumvent potential
ground effect aerodynamics at low simulated descent rates. The rotors
could be easily interchanged and were individually driven by a brush-
less DC motor, which was controlled by a pulse-width modulation
(PWM) signal generated by a microcontroller. A consistent power
input was provided by a 12 VDC power supply. The setup was further
instrumented with a power meter and an electrical revolutions per
minute (RPM) counter, which measured the rotational speed of the
brushless motor via voltage changes in the wires between the electronic
speed controller (ESC) and the motor. The correctness of readings was
verified using a handheld optical RPM sensor prior to conducting the
aerodynamic wind tunnel tests. Since the wind tunnel setup lacked a
torque sensor, an extensive hover characterization of all rotor blades
was performed on a dedicated
RC Benchmark
Series 1580 thrust stand
for more precise and conclusive identification of rotor constants (e.g.,
C
T
,
C
P
,and
FM
).
B. Experimental procedure
The testing strategy for establishing the aerodynamic descent
characteristics for each rotor blade design and examining the effect of
rotor geometry was based on constant rotor rotation rate throughout a
test run and quasi-steady-state descent operation and was as follows:
with the wind tunnel at rest, predetermined PWM signals were sent to
the motor, spooling up to a corresponding rotation rate. The generated
mean rotor thrust was measured over a 10 s interval, equating to the
thrust of the rotor in hover,
T
h
(i.e., no external freestream flow).
Simultaneously, the standard deviation of the thrust was determined
over the same interval. Subsequently, while maintaining a constant
rotor rotation rate, the wind tunnel velocity was incrementally
increased in approximately 0.5 m/s-increments up to 9.6 m/s to simu-
late increasing descent rates. For each velocity increment, the gener-
ated thrust force was measured over a 10 s period, and the mean and
standard deviation of the thrust were quantified. The notation used
from here on is that
T
0
represents the continuous thrust measure-
mentsovertheinterval,and
T
denotes the average [
T
ð
v
c
Þ¼
T
0
ð
v
c
Þ
and
T
h
¼
T
ð
0
Þ¼
T
0
ð
0
Þ
]. For each rotor design, this test procedure
was repeated for a series of rotation rates corresponding to predefined
PWM-motor-inputs starting from 1200 up to 1400
l
sin50
l
ssteps,
which resulted in rotation rates ranging from around 5000 RPM to
10 000 RPM. To facilitate comparisons between data series, the mea-
sured mean thrust,
T
, was normalized by the hover thrust,
T
h
to iden-
tify the relative thrust loss compared to hover conditions for constant
rotation rates as a function of descent velocities. Meanwhile, the simu-
lated descent velocity (i.e., the wind tunnel freestream velocity),
v
c
,was
normalized by the equivalent induced velocity in hover,
v
h
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T
h
=
2
q
A
p
. Parasitic drag forces acting on the supporting struc-
ture of the rotor assembly were taken into account and were subtracted
from the raw mean aerodynamic force measurements. For this, drag
baselines were established by measuring the generated forces of the
rotor assembly after removing the rotor blade (including motor, con-
nection tube to the load cell, wiring). The unsteady contribution of the
drag baselines was determined to be negligible compared to the fluctua-
tions measured for the rotor tests and was, therefore, omitted.
Figure 7
provides an illustration of the experimental procedure by plotting the
raw and normalized mean thrust measurements for the reference blade.
As can be seen, using different rotation rates (and, therefore, thrust lev-
els) across the test runs helps one to explore the effects of varying the
thrust level and tip Reynolds number on the descent performance.
Simultaneously, different ratios of
v
c
=
v
h
can be obtained for pre-
specified values of
v
c
when varying the initial hover thrust,
T
h
, for more
accurate identification of the critical descent rate ratio and associated
maximum thrust losses. Exact results and findings regarding the
descent behavior are described in more detail in Sec.
V
.
Analogous to the mean thrust, the thrust fluctuations were exam-
ined as a function of the descent rate. Since VRS-induced fluctuations
are characteristically low-frequency fluctuations
1,5
and were reported
to be on the order of multiple rotor rotations,
4
the recorded thrust his-
tory was low-pass filtered to 50 Hz, lower than the rotor lowest rotor
rotational frequency to isolate the fluctuations induced by the VRS
behavior from the rotational frequencies. The measured standard devi-
ation of the low-pass filtered thrust for each velocity increment,
SD
ð
T
0
Þ
, was also normalized by the arithmetic mean,
T
,overthesame
interval. Note that the standard deviation is denoted here as “
SD
”
instead of
r
to avoid confusion with the rotor solidity.
C. PIV setup
In addition to aerodynamic force measurements, flow visualiza-
tion was performed in a horizontal water tunnel using a two-
dimensional PIV setup. The PIV setup is schematically shown in
Fig. 5
. A water tunnel was selected for flow visualizations, as opposed
to a wind tunnel, because rotational speeds and flow velocities at com-
parable Reynolds numbers were significantly reduced, allowing us to
better capture the dynamic behavior of the highly time dependent, dis-
ordered flow field. Throughout the PIV study, particular interest was
dedicated toward the formation and trajectories of the trailed rotor tip
vortices. All 3D-printed rotor blade geometries were investigated
FIG. 5.
Schematic illustration of the PIV setup.
Physics of Fluids
ARTICLE
scitation.org/journal/phf
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
34
, 035124-7
Published under an exclusive license by AIP Publishing
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under hover conditions (no external flow) for comparison purposes,
and the reference blade of this study was furthermore subjected to a
counter-flow to simulate descent conditions. All rotors operated fully
submerged in the horizontal water tunnel at identical PWM-motor-
inputs resulting in rotational speeds of approximately five rotations per
second. The flow field was illuminated with a continuous 2D laser sheet
and imaged using a high speed camera at 1000 fps, resulting
D
t
¼
1ms
(
D
w
¼
1
:
8
) between two consecutive images. Ensemble averaged
(2000 frames) and phase averaged (500 frames) data sets were collected
and subsequently processed using a PIV software for every rotor. The
raw image files recorded by the camera had a resolution of 1920
1080
pixels (
10 px/mm). Using an 8
8 pixel interrogation area and 50%
overlap, the obtained velocity fields had a spatial resolution of 479
269 vectors (
2.5 vectors/mm). Multiple interrogation area sizes and
overlap settings were evaluated with only minimal differences between
results. Phase averaging was enabled by a magnetic trigger, which sent
an electric trigger pulse to the camera to record sets of two images, sepa-
rated by
D
t
¼
1 ms (1000 fps) when the rotor assumed a pre-specified
azimuth angle. In total, four phased averaged data sets at azimuth angles
relative to the laser sheet (
w
¼
15
;
30
;
45
;
and 90
) were recorded
for each rotor. Meanwhile, the ensemble average data sets included 2000
images, continuously recorded at 1000 fps (
D
t
¼
1ms
;
D
w
¼
1
:
8
)up
to
w
max
¼
3600
.
Figure 6
is provided to illustrate the PIV acquisition
and processing procedure. Because no equipment for force measure-
ments under water was available, the generated thrust [via Eq.
(4)
]and
theoretical induced velocity in hover were calculated by
T
h
¼
q
w
C
T
p
R
2
ð
X
R
Þ
2
;
(11)
v
h
¼
ffiffiffiffiffiffiffiffiffiffiffi
T
h
2
q
w
A
s
;
(12)
where
q
w
is the density of water. As previously mentioned,
C
T
-
coefficients for each rotor geometry were separately established using
the
RC Benchmark
rotor test stand. These
C
T
-coefficients should be
representative, as rotors operated under similar Reynolds numbers in
air and water. The flow visualization proved to be a useful resource in
confirming a tendency of the tip vortices to remain within the rotor
plane in the VRS and providing insight on how the unsteadiness in the
disk loading can be reduced by systematic tip vortex control.
V. DISCUSSION OF RESULTS
A. Wind tunnel experiments
To examine how geometric parameters influence the axial
descent performance of a rotor, the mean thrust and thrust standard
deviation were evaluated as a function of the descent rate ratio for all
modifications of the rectangular reference blade listed in
Table III
by
following the experimental procedure described in Sec.
IV B
.
Figure 7
, showing the mean thrust measurements of the reference
blade (
Ref
) against the climb velocity,
v
c
, helps one to illustrate the
experimental procedure. A total of five data sets were collected, corre-
sponding to five separate test runs at their respective, constant RPM,
with each data point denoting a 10 s thrust average at a given climb
velocity. Note that climb velocities are negative, which are why descent
rates increase from right to left in the plots. The data normalization in
Fig. 7
(top right) brings all measurements into alignment to a single,
characteristic curve. These results indicate that for a constant rotation
FIG. 6.
Schematic illustration PIV data acquisition and processing procedure.
Physics of Fluids
ARTICLE
scitation.org/journal/phf
Phys. Fluids
34
, 035124 (2022); doi: 10.1063/5.0083761
34
, 035124-8
Published under an exclusive license by AIP Publishing
06 October 2023 18:43:59