counter rotating propeller design using blade …
TRANSCRIPT
1
ISABE-2015-20196
COUNTER ROTATING PROPELLER DESIGN USING BLADE ELEMENT MOMENTUM THEORY
Kiran Siddappaji and Mark G. Turner
Gas Turbine Simulation Laboratory
University of Cincinnati
Cincinnati, Ohio 45221 USA
Abstract
Counter rotating propellers (CRProp) have many
advantages over single propeller systems. Rotational
energy lost in the slipstream is recovered by the aft
rotor, smaller diameter and lower loading per blade
and total torque is balanced due to the opposite
rotation of the rotors. A design method is presented
using Blade Element Momentum Theory (BEMT)
combining Blade Momentum (BMT) and Blade
Element (BET) theories. The tip loss effect
(Prandtl's) and the wake rotation due to the front rotor
is also accounted. The non-linear relationship
between the flow angle and the axial and angular
induction factors is converted to a linear equation and
is solved using Brent’s method with appropriate
boundary conditions. The axial gap between the
rotors is also accounted in all the calculations.
Spanwise distribution of chord, stagger angles and
airfoil type is utilized as initial geometry and a 3D
blade shape is created using an in-house 3D geometry
generator (3DBGB). Appropriate airfoil lift and drag
properties are obtained from a look-up table as a
function of angle of attack for a wide range of
Reynolds number (REYN) as it varies spanwise. A
design optimization is performed with a single
objective function using genetic algorithm to obtain a
blade design with maximum efficiency for the
counter rotating propeller configuration for a
specified thrust. The robust implementation of
Brent’s method and BEMT makes the design process
fast, computationally cheaper and quicker
optimization cycles.
Nomenclature
A Intermediate term for a' calculation
a Axial Induction factor
a' Angular Induction factor
C Coefficient of lift/drag/axial/normal forces
c(r) Chord at a radial location
dQ Incremental Torque
dr Incremental radius
dT Incremental Thrust
dP Incremental Power
e Mathematical Constant: 2.71828
F Total Loss factor, Axial Force
f Prandtl loss factor
n Revolutions per second
P Power
Q Torque
R Radius at hub and tip
r Radius at any location
T Thrust
V Velocity of the fluid
x Rotor axial gap coefficient
Z Blade count
Greek Symbols
α Angle of Attack
κ Intermediate term for a calculation
λ Tip speed ratio
Ω Rotor rotational speed
ω Wake rotational speed
φ Flow Angle
π Circumference to diameter ratio for circle
ρ Fluid density
σ Blade solidity
θ Twist angle
Subscripts
1 Rotor 1 or Front Rotor
1-7 Axial stations 1 to 7
12 Influence on rotor 2 due to rotor 1
2 Rotor 2 or Aft Rotor
a Axial
d Drag
l Lift
local Local twist
n Normal
p Propeller
r Radial
total Total
z Axial direction or Free stream
2
Abbreviations
3D Three Dimensional
3DBGB 3D Blade Geometry Builder
AoA Angle of Attack
BEMT Blade Element Momentum Theory
BET Blade Element Theory
BMT Blade Momentum Theory
CFD Computational Fluid Dynamics
CRProp Counter Rotating Propeller
REYN Reynolds Number
RPM Revolutions Per Minute
TSR Tip Speed Ratio
Introduction
Counter rotating propeller systems have a front and
aft propeller spinning in opposite directions. The exit
swirl coming out of the front rotor is utilized by the
aft rotor and the rotational energy which is normally
lost in the slipstream is recovered. The other benefits
of this configuration are torque balance due to the
counter rotation, higher efficiency for a given disk
area due to lower loading per blade and smaller
optimum diameter. The challenges are the complex
shafting and gear systems. The electrical propulsion
systems have made it easier for application of these
devices in multicopter. Multicopters are beginning to
gain popularity in a variety of fields and an efficient
propulsion system is essential along with smart
control systems for adopting this technology.
There are several propeller theories which analyze
the aerodynamic performance and the behavior of the
slipstream. Lock et al. (1941) developed a method for
calculating the performance of a pair of contra-
rotating airscrews including the interference
velocities between the rotors through a blade force
analysis with some assumptions and appropriate
velocity diagram. Chen et al. (1990) developed a
design system for contra-rotating propellers using
momentum, mass and circulation conservation
through lifting-line theory for calculating propeller
loadings and rotor interaction velocities. Three
dimensional viscous flow models are also
implemented but cannot be included in an iterative
geometry manipulation process because of high
computational cost and longer solution time. A low
fidelity approach is widely incorporated in the design
process which uses a combination (BEMT) of blade
momentum theory (BMT), a derivative of actuator
disk theory to relate the flow momentum at the
propeller to that of the far wake and the blade
element theory to predict the propeller forces (M.K.
Rwigema, 2010). Ernesto Benini (2004) implemented
the combined momentum-blade element theory for
marine propellers using a panel/integral boundary
layer method for calculating the properties of each
blade profile. Furthermore, he demonstrated that the
performance prediction with this approach is accurate
for a certain range of the advance ratio and deviates
from the 3D CFD results outside this range due to 3D
effects. Hence, application of BEMT is simple, fast,
computationally inexpensive and accurate within a
certain range.
QPROP and XROTOR (Mark Drela, 2006) are
design and analysis programs for predicting the
performance of propeller-motor combinations and
minimum induced loss rotor for free-tip propellers
along with structural and acoustic analysis
respectively which use an extension of the classical
blade-element/vortex formulation. BEMT is used to
calculate the axial and angular induction factors in
order to calculate the spanwise distribution of
geometric and aerodynamic properties. Typically,
numerical methods used to solve the equations to
calculate the induction factors are either fixed point
iteration or variations of newton’s method. Ning
(2013) converted the non-linear relationship between
the induction factors and the flow angle to a linear
equation and used Brent’s method for obtaining the
flow angles with guaranteed convergence
implementing proper boundary conditions.
The assumptions made in BMT are homogenous;
incompressible; steady state flow; no frictional drag;
uniform thrust over the ideal propeller (disk); a non-
rotating wake and the static pressure far upstream and
downstream is equal to the ambient static pressure.
BET divides the rotor blade into discrete span-wise
elements with an incremental radius and assumes no
fluid interaction between the airfoil elements which
makes the flow completely two dimensional. The lift
and drag properties of the airfoil are usually obtained
by running XFOIL (Mark Drela, 2013) on the chosen
geometry at a specific Reynolds number for a chosen
angle of attack.
3
Application
An unducted CRProp is designed for a multicopter
with 150 pounds (667.24 N) of thrust per pod,
suitable for rescue and relief operations in natural
disaster areas. A 10 cm hub diameter is assumed. The
number of pods for the multicopter can be chosen
based on the total thrust requirement.
Methodology
A process flowchart describing the design steps are
shown in Figure 1.
Figure 1: Process Flowchart for CRProp design
using BEMT.
Airfoil Selection
The lift and drag properties of an airfoil depend on
the Reynolds number and the angle of attack. Clark Y
airfoil was chosen with 11.71% thickness, 28%
maximum thickness position, 3.43% maximum
camber and 42% maximum camber position for the
design and analysis. A vast library of the airfoil lift
and drag properties as a function of angle of attack
(-10.00 to 10.00 degrees) for a wide range of
Reynolds number (10.0E4 – 11.6E6) is created using
XFLR5 (Deperrois, A., 2009) as shown in Figure 2.
This library is used as a look-up table to obtain the
airfoil properties spanwise at corresponding Reynolds
number. The BEMT code is a unified approach for
both wind or hydrokinetic turbines and propellers. In
order to use the same set of equations for both cases,
the chosen convention for propeller is negative
induction factors, thrust, power and angle of attack
whereas for turbines they are taken as positive. The
Cl-AoA curve shown in Figure 2 is taken as a mirror
image about the origin to obtain lift and drag
coefficients for negative angle of attack since a
propeller blade is an upside down wind turbine blade.
Figure 2: Airfoil lift and drag coefficients for
Clark Y airfoil for a range of REYN (10.0E4 –
11.6E6) as a function of angle of attacks (-10.00 to
10.00) using XFLR5 (Deperrois, A., 2009).
tiphubtotal
tip
tip
hubhub
f
FFF
r
rRZf
r
RrZf
eF
sin2
)(
sin2
)(
)(cos2 1
(1)
Blade Element Momentum Theory
The CRProp configuration and the streamtubes are
shown in Figure 3. The streamtube contraction is
exactly opposite to the streamtube expansion in a
wind turbine case as shown by Sairam et al. (2014).
The axial gap between the 2 rotors also affects the
performance as shown by Pundhir et al. (1992). It is
assumed that the 2 rotors are very close to each other.
Tip losses are accounted by using a Prandtl tip loss
4
factor (Shen et al., 2005) as shown in equation (1).
Applying Bernoulli’s principle upstream and
downstream of the front rotor, we get
)(2
1
2
1
2
1
2
1
2
1
2
4
2
132
2
44
2
33
2
22
2
11
VVPP
VPVP
VPVP
ss
ss
ss
(2)
The mass flow rate is conserved in the streamtube
and the axial force (thrust) is given as
)()(2
1
)()(
)(
)(
22
4
2
1
2
32
41
2
2
rVV
rPPThrust
VVmF
rVm
ss
A
(3)
)1(
)1(
0
2
)(2
1)(
114
23
112
1
12
1
21
1
41
2
2
4
2
141
xaVV
VV
aVV
a
VV
V
VVa
VVV
VVVVm
(4)
Equating the axial force and the thrust, an expression
for the velocity at the rotor is obtained. Axial
induction factor is defined as the ratio of the
difference of free stream velocity and velocity at the
rotor to the free stream velocity as shown in equation
(4) and is negative because the velocity increases
axially due to the streamtube contraction. Velocities
at other axial locations are also calculated and can be
extended to the aft rotor as well. x is the rotor axial
gap coefficient ranging from 1 to 2.
Blade momentum and blade element theories are
combined to calculate the induction factors and other
performance parameters. The velocity triangles are
shown in Figure 4. Angular induction factor is the
ratio of the wake rotational speed to the rotor
rotational speed as shown in equation (5). Axial and
angular momentum conservation across the rotors
give the incremental thrust and torque for the front
(dT1, dQ1) and aft rotor (dT2, dQ2) as shown in
equation (5), where Ftotal is the total loss factor.
drrVaaaFdQ
rdrVaaFdT
drrVaaFdQ
rrardrVdQ
rrmddQ
a
rdrVxaxaFdT
rdrVxaxadT
rdrPPdT
total
total
ztotal
ztotal
z
ss
3
2421222
2
4222
3
1111
121
1
1
11
22
111
22
111
321
)1)(''(4
)1(4
)1('4
)'2(2
)(
2'
)2(4
)2(4
2)(
(5)
In equation (6), a’12 is the angular induction factor
due to the wake rotation at the inlet of the aft rotor
exiting from the front rotor, ω12 is the wake rotation
speed and Ω2 is the aft rotor rotation speed.
Figure 3: CRProp configuration and axial
velocities at different axial stations. (Sairam et al,
2014)
)'1(')1(
4112
1'
2'
222
2
22
12
2
1212
aaaa
A
Aa
a
r
p
p
(6)
Similarly, according to the BET for the front rotor,
we have,
5
z
tip
tip
r
r
dln
dla
nz
az
V
R
R
r
a
a
CCC
CCC
drrCaV
dQ
rdrCaV
dT
11
1
1
11
11
11
11111
11111
2
1
1
2
2
1
2
11
1
1
2
2
1
2
11
)'1(
1tan
cossin
sincos
sin
)1(
sin
)1(
(7)
and for the aft rotor,
4
22
2
2
22
22
22
22222
22222
2
2
2
2
2
2
2
422
2
2
2
2
2
2
422
)'1(
1tan
cossin
sincos
sin
)1(
sin
)1(
V
R
R
r
a
a
CCC
CCC
drrCaV
dQ
rdrCaV
dT
tip
tip
r
r
dln
dla
n
a
(8)
where,
222
111
14
]2,1[
)1(
local
local
z
x
xaVV
(9)
Figure 4: Velocity triangles for CRProp. (Su et al,
2012)
σ1, σ2 are the solidities for respective rotors. The
relationship between advance ratio and the tip speed
ratio is given in equation (10).
J
nD
VJ
tip
z
(10)
Spanwise definition of chord, AoA and REYN
The spanwise chord is given and can be scaled
smoothly using chord multipliers defined as several
control points which create a smooth parametric
cubic B-spline (Vince, J., 2006). The angle of attack
can also be varied spanwise using B-spline control
points. The Reynolds number varies across the blade
span and the corresponding airfoil properties can be
obtained using a REYN based library of Cl and Cd
values for a range of AoAs.
Calculation of Induction Factors
The relationship between axial and angular induction
factors and the flow angle is non-linear and reducing
it to a linear equation is very beneficial in terms of
numerical convergence and obtaining a unique
solution. The reduced single equations for both rotors
are shown in equations (11) and (12).
0)'1(
cos)1(sin
0)'1(
cos
1
sin
2
22
11
1
1
1
r
r aa
(11)
22
22
2
2
22
cossin4'
sin4
total
n
total
a
F
C
F
C
(12)
The roots of equation (11) are obtained using Brent’s
method with proper boundary conditions (Brent,
1971 and Ning, 2013). The relationship of a and a’ is
given in equation (13) and (14) for the front and aft
rotors. The axial gap coefficient (x) affects the axial
induction factor for the front rotor. The spanwise
distribution of incremental thrust, torque, power,
flow, metal and stagger angles are calculated once all
6
the necessary variables are obtained. The total thrust
and the torque is the summation of the dT and dQ for
all elemental airfoils spanwise and the power is
obtained as shown in equation (15).
1cossin4
1'
sin4
1
)1()1(1
11
11
1
11
1
2
2
22
1
n
total
a
total
p
p
ppp
C
Fa
C
F
x
xxxa
(13)
1cossin4
1'
1sin4
1
22
222
22
2
22
n
total
a
total
C
Fa
C
Fa
(14)
tip
hub
tip
hub
tip
hub
R
R
R
R
R
R
drdr
dQP
dQdP
drdr
dTT
drdr
dQQ
)(
)(
)(
(15)
The coefficient of power, thrust and the CRProp
efficiency are calculated using equation (16).
AftFront
zAftFront
opCR
P
T
tip
zop
tip
T
tip
P
PP
VTT
C
C
Rn
V
Rn
TC
Rn
PC
)(
)2(
)2(
)2(
Pr
Pr
42
53
(16)
3D Parametric Geometry
The spanwise chord, stagger angles and the airfoil
type obtained from the BEMT code is used to create
a 3D blade geometry using an in-house parametric
3D blade geometry builder (Siddappaji et al., 2012).
A variety of parameters can be modified in 3DBGB
to create blade shapes for a wide range of operating
conditions. Several types of airfoils can be imported
at different spanwise locations. This 3D geometry
serves as an initial step in 3D CFD analysis,
structural analysis and design optimization, hence
bridging the low fidelity with the high fidelity
system. Figure 5 shows a 3D geometry of a counter
rotating propeller model using the initial parameters
from the BEMT code.
Figure 5: Counter rotating propeller 3D model for
the baseline design with constant REYN.
Design Optimization using BEMT code
The optimization was performed for maximum
efficiency as a single objective function using
DAKOTA (http://dakota.sandia.gov.) and the BEMT
code for the counter rotating propeller.
Genetic Algorithm was utilized which is based on
the natural selection process and evolution in nature.
Single-Objective Genetic Algorithm (SOGA) from
the John Eddy Genetic Algorithm (JEGA) library of
DAKOTA is used in the optimization. DAKOTA
controls the optimization as shown in Figure 6. The
optimization is done in serial due to the fast
convergence of the BEMT code and the details of the
DAKOTA optimization process is explained by
Siddappaji et al. (2015).
Figure 6: DAKOTA Optimization flowchart.
7
Optimization parameters
The goal was to obtain a design with optimum
efficiency. The single objective function to be
minimized was set to (1-η). No mechanical constraint
was enforced in order to obtain the universal
optimum and to understand the behavior of the
design parameters. Fixed design parameters for
counter rotating propeller design optimization are
1. Reynolds Number = 50000 ; α = -4°
2. Airfoil type = Clark Y ; Vz = 15 m/s
3. Rhub = 0.05 m
4. Number of spanwise airfoil sections : 21
5. Axial gap between the rotors.
Variable design parameters (total 16) for the counter
rotating propeller are given in Table 1.
Variables Front Aft
Rtip [m] 0.55-0.85 0.45-0.85
Z 3-6 3-6
TSR, RPM 7.0-12.0 -3500 to -1000
5 chord
multipliers
0-75% 1.0-3.0
>75% 1.0-4.0
0-75% 1.0-4.0
>75% 1.0-5.0
Table 1: Varying parameters for the CRProp
optimization.
Results
Air is used as the fluid with a density of 1.225 kg/m3,
kinematic viscosity of 15.68E-6 m2/s at 20
°C.
Constant REYN and constant AoA
The baseline design details are given in Table 2. The
optimization was performed by varying 16
parameters using DAKOTA which took 14 minutes
and 8 seconds to be completed using 1438 function
evaluations on a single core of an Intel I7 processor.
The optimized design has an improved total
efficiency of 66.49% from the baseline efficiency of
53.09% with a total thrust of 166.25 pounds.
Furthermore, the front rotor is 3 bladed and the
efficiency is improved to 67.40% from 60.09% and
the aft rotor is 4 bladed and the efficiency is
improved to 65.42% from 47.82%. The optimized
design characteristics are tabulated in Table 3. Figure
7a shows the spanwise distribution of induction
factors and the velocity distribution at various axial
locations for the baseline design cases of constant
and varying REYN. The power and thrust
coefficients for a range of advance ratios for both
baseline cases are shown in Figure 7b. Figure 10
shows the spanwise distribution of chord, stagger
angles for both baseline and optimized cases in
addition to the induction factors and axial velocities.
a1, a’1 are the axial and angular induction factors for
the front rotor and a2, a’2 for the aft rotor; a’12 is the
angular induction factor due to the exit wake from the
front rotor interacting with the aft rotor. The span is
calculated with respect to the front rotor tip radius
which results in shorter spans for the aft rotor as
shown in the radial plots. Vz_R1 is the inlet velocity
for the front rotor; V_LE_R1 is the velocity at the
front rotor leading edge and is equal to V_TE_R1,
velocity at the front rotor trailing edge. Vz_R2 is the
inlet velocity for the aft rotor and is equal to the
velocity at the front rotor trailing edge as the 2 rotors
are very close to each other.
Properties Units FRONT AFT
Hub Dia. cm 10.00 10.00
Tip Dia. cm 130.00 127.50
Vz m/s 15.0 16.79-19.03
Z - 3 3
TSR - 9.18 -8.26
RPM - 2023.20 -2300.00
Chord m specified specified
Thrust N
(lbf)
-335.80
(-75.49)
-354.70
(-79.74)
Torque Nm -39.56 46.19
Power kW -8.38 -11.13
Rotor Eff. % 60.09 47.82
ηCRProp % 53.09
Table 2: Design properties of the baseline CRProp
design for constant REYN spanwise.
Properties Units FRONT AFT
Hub Dia. cm 10.00 10.00
Tip Dia. cm 165.61 91.55
Vz m/s 15.0 15.71-18.56
Z - 3 4
TSR - 7.06 -3.23
RPM - 1220.88 -1223.36
Chord m specified specified
Thrust N
(lbf)
-403.21
(-90.65)
-336.30
(-75.60)
Torque Nm -70.19 60.19
Power kW -8.97 -7.71
Rotor Eff. % 67.40 65.42
ηCRProp % 66.49
Table 3: Optimized design properties with
constant REYN along the span.
8
Figure 7a: a, a’ and Vz for constant (left) and varying REYN (right) baseline cases.
Figure 7b: Thrust (top) and power (bottom) coefficients for a range of advance ratio for constant (left) and
varying REYN (right) baseline cases.
9
It can be clearly seen that the inlet velocity for the
aft rotor is no longer a constant. V_LE_R2 is the
velocity at the aft rotor leading edge and is equal to
the velocity at the aft rotor trailing edge, V_TE_R2.
Vexit is the exit velocity.
Varying REYN and constant AoA
The Reynolds number varies along the blade span for
both rotors as shown in Figure 8a for the baseline
case and the baseline design details are shown in
Table 4. In this optimization, appropriate lift and drag
coefficients were calculated along the span, obtained
from a look-up table of the coefficients and Reynolds
number for a constant AoA. The Reynolds number
variation for the optimized case is shown in Figure 8b
for both front and aft rotors. The optimization was
performed by varying 16 parameters using DAKOTA
which took 26 minutes and 8 seconds to be
completed using 2502 function evaluations on a
single core of an Intel I7 processor.
Properties Units FRONT AFT
Hub Dia. cm 10.00 10.00
Tip Dia. cm 130.00 110.00
Vz m/s 15.0 16.63-19.52
Z - 3 3
TSR - 9.18 -7.61
RPM - 2023.2 -2200.00
Chord m specified specified
Thrust N
(lbf)
-386.32
(-86.85)
-266.30
(-59.87)
Torque Nm -42.05 31.82
Power kW -8.91 -7.33
Rotor Eff. % 65.04 54.49
ηCRProp % 60.28
Table 4: Design properties of the baseline CRProp
design for REYN varying spanwise.
The optimized design has an improved total
efficiency of 73.87% from the baseline efficiency of
60.28% with a total thrust of 153.03 pounds.
Furthermore, the front rotor is 3 bladed and the
efficiency is improved to 72.18% from 65.04% and
the aft rotor is 5 bladed and the efficiency is
improved to 75.75% from 54.49%. The optimized
design characteristics are tabulated in Table 5. Figure
10 shows the comparison of spanwise distribution of
chord, stagger, induction factors and axial velocities
at various axial locations for the varying REYN and
constant REYN optimization cases. The AoA is
constant for both cases. The 3D geometries for both
optimum cases are shown in Figure 9.
Figure 8a: Spanwise variation of REYN (10
5) for
the baseline case.
Figure 8b: Spanwise variation of REYN (10
5) for
the optimized case.
Properties Units FRONT AFT
Hub Dia. cm 10.00 10.00
Tip Dia. cm 142.87 94.54
Vz m/s 15.0 16.49-18.82
Z - 3 5
TSR - 7.05 -2.88
RPM - 1413.25 -1069.03
Chord m specified specified
Thrust N
(lbf)
-351.04
(-78.92)
-329.68
(-74.11)
Torque Nm -49.29 58.32
Power kW -7.29 -6.53
Rotor Eff. % 72.18 75.75
ηCRProp % 73.87
Table 5: Optimized design properties with REYN
varying along the span.
10
Conclusions and Future Work
A robust design system for counter rotating
propellers is created using blade element momentum
theory. Counter rotating propellers are advantageous
over single row propellers due to the fact that an
additional amount of power is extracted from the
front rotor exit swirl by the counter rotating aft rotor.
The axial gap between the rotors is also accounted in
the formulation and the rotors are assumed to be very
close to each other. The low fidelity design tool takes
into account the tip losses using a Prandtl’s tip loss
model. The non-linear relationship between induction
factors and flow angle is linearized and solved using
Brent’s method with appropriate boundary conditions
which ensures convergence. Appropriate airfoil lift
and drag properties are obtained as a function of
AoAs for a wide range of Reynolds number. The
design is also optimized for maximum total
efficiency using genetic algorithm. The optimization
was performed with no mechanical constraints to
study the optimum design space.
Future work includes optimization with varying
AoA and airfoil spanwise, 3D CFD and structural
analysis. A suitable wake model needs to be added to
account for wakes. Entropy minimization is essential
to improve the efficiency. Blade-to-blade interaction
is necessary for designs with more blades. Design
exploration with swept and leaned blades, novel
features like rotor winglets and split tips will be
conducted. A study of radius ratios and nose effect on
velocity at the hub for smaller size propellers will
also be investigated.
In conclusion, the robust implementation of Brent’s
method and BEMT makes the design process fast,
computationally cheaper and quicker optimization
cycles. It is demonstrated that BEMT combined with
optimization is a great tool for obtaining efficient
designs rapidly for counter rotating propellers and
when coupled with a parametric 3D blade geometry
tool, it can function as a preprocess to high fidelity
design and analysis.
Figure 9: 3D geometry lofted in a CAD package (NX) for the CRProp optimized design case with constant
(top) and varying REYN (bottom).
11
Figure 10: Comparison of spanwise distribution of chord, stagger, induction factors and Vz at various axial
locations for the constant REYN (left) and varying REYN (right) optimization cases of CRProp.
12
References
Lock, C. N. H., July 1941. “Interference velocity for
a close pair of contra-rotating airscrews”, Tech. Rep.
No. 2084, Aerodynamics Division, NPL, London,
UK.
Chen, B. Y. H., and Reed, A. M., January 1990. “A
design method and an application for contra rotating
propellers”, Tech. Rep. AD-A218625, David Taylor
Research Center, Bethesda, MD 20084-5000,
Rwigema, M. K., September 2010, “Propeller Blade
Element Momentum Theory with Vortex Wake
Deflection”, 27th
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