48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br>15th
23 - 26 April 2007, Honolulu, Hawaii
AIAA 2007-1883
AIAA-2007-1883
rd
3 AIAA Multidisciplinary Design Optimization Specialist Conference
23-26 April 2007, Honolulu, Hawaii
Performance and Noise Optimization of a Propeller using the
Vortex Lattice Method and a Genetic Algorithm
Christoph Burger°, Roy Hartfield†, John Burkhalter‡
Aerospace Engineering, Auburn University, AL
Abstract
This paper examines the viability of propeller design using the vortex lattice
method for performance prediction, an empirical model for propeller noise
determination and a genetic algorithm for the design optimization. This paper
includes a detailed description of the propeller geometry scheme which is based on a
universal parametric geometry representation method developed by Kulfan. Results
include single and multipoint optimization efforts with and without compressibility
considerations.
Introduction
In recent years fixed wing and rotary wing UAV’s have become a viable tool not only
for military reconnaissance and attack missions, but also for commercial applications
such as aerial photography, forest fire assessment, weather observations and border
patrol. In military applications close urban reconnaissance requires a silent propulsion
system to avoid detection by the foe. This investigation is concerned with the
multidisciplinary performance optimization of a propeller with minimal noise signature.
In this effort, the aerodynamic principles of lifting surface theory as developed using
vortex lattice techniques and noise prediction using an empirical analytic method are used
to evaluate the overall performance of candidate propeller combinations. A genetic
algorithm (GA) optimization scheme is used to drive the design. This general approach
is adapted from the aircraft propeller and wind turbine optimization efforts described in
Refs.1 and 2.
º
Graduate Student, Aerospace Eng. Dep., 211 Aerospace Eng. Bld. Auburn, AL, 36849, Member.
Associate Professor, Aerospace Eng. Dep., 211 Aerospace Eng. Bld., Auburn, AL, 36849, Senior
Member.
‡
Professor Emeritus, Aerospace Eng. Dep., 211 Aerospace Eng. Bld, Auburn, AL, 36849, Senior Member.
†
1
Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Background
Even though the vortex lattice theory does not directly account for any viscous or
compressibility effects, it has been shown that propeller performance predictions using
this method can be close to experimental performance data. Masquelier3 applied the
Vortex Lattice Method (VLM) to predict the performance of non chambered constant
chord propellers at different angles of attack and advance ratios. Results were compared
to performance data obtained by blade element theory. Kobayakawa4 predicted the
performance of an advanced turboprop using the vortex lattice method and found good
agreement with experimental data. Compressibility effects can be included through the
Prandtl-Glauert similarity rule. Cheung5 showed that thin airfoil propeller performance is
predicted well with a single layer of vortex elements when compared to experimental
data. For propellers with thick airfoils two layers of vortex elements are required to
accurately predict aerodynamic propeller performance. The propeller geometry definition
used in this effort is based on Bernstein polynomials which describe general geometric
shapes efficiently.
For the optimization the Improve© Code GA developed and implemented in several
aerospace applications by Anderson6-8 was used to drive the propeller geometry. The GA
uses a tournament based method for reproduction and has options for elitism and pareto
development of multiple goal problems. The parameters of interest were propeller power,
torque and far field noise. Optimizations include single and dual wind and propeller
rotational speed optimizations. The GA was chosen as the optimizer because some
parameters (number of blades for example) are discrete, eliminating the use of gradient
methods and because the GA is very efficient as compared to other population based
methods.
Propeller Performance Program
General
The propeller performance program is the principle portion of the objective function
for the GA. The program consists of four major components including: A routine to
generate the airfoil and propeller geometry, a routine for the grid discretization, a routine
to determine the influence coefficients and the Gauss Seidel solver subroutine to solve for
the unknown circulations. The geometry of the propeller is defined by 66 variables, 48
describe the upper and lower airfoil geometry while the remaining 18 describe the angle
off attack function, the spanwise chord length variation, the propeller sweep function, the
number of propeller blades and the propeller radius. The 66 propeller geometry design
variables are allowed to change during the optimization process and other fixed
parameters such as the free stream velocity, rotational speed, number of panels and
propeller hub diameter are constants.
Propeller Geometry Generation
Several methods have been developed to represent general airfoil shapes for use in
aerodynamic design optimization. The goal is to define a simple analytic function which
efficiently describes the entire design space for airfoils. Some of the methods used in
previous efforts can be found in Ref. 9-14.
2
The method chosen is based on the work of Ref.15 and Ref.16. In this approach a
simple and well behaved analytic unit shape function, based on Bernstein polynomials is
introduced. This shape function directly controls key airfoil parameters including leading
edge radius, thickness and trailing edge angle. A unit class function is added to the unit
shape function by multiplying it with the shape function to allow for a wider variety of
general body shapes. The two terms in the class function enforce specific shapes at the
leading and trailing edge, while influencing the center section shapes minor.
The streamwise class function in the design space is defined as
C NN21 (ψ ) = (ψ ) 1 ⋅ [1 − ψ ]
N
N2
(1)
with ψ being the fraction of the local chord. In the physical space the unit class function
is
⎛ x⎞ ⎛ x⎞
C⎜ ⎟ = ⎜ ⎟
⎝c⎠ ⎝c⎠
N1
⎡
⋅ ⎢1 −
⎣
x⎤
c ⎥⎦
N2
(2)
with the chord c and x the local variable with a range from 0-1.0. The first term of
equation (2) defines the shape of the airfoil leading edge and the second term can be used
to ensure a sharp trailing edge. If N1 = 0.5 and N2 = 1.0 a round airfoil leading edge and a
sharp trailing is enforced. Further geometry class determinations due to N1 and N2
variation can be found in Ref.15.
The unit shape function is defined by a Bernstein polynomial of the order n with the
variable x ranging from 0-1.0 and r from 0 to n. The Bernstein polynomials were chosen
due to the mathematical property of “Partition of Unity” as described in Ref.16. The first
term of the shape function defines the binomial coefficients with increasing order n of the
BP. For each order n of BP there exist n+1 terms which are defined by equation 3.
n!
⎛ x⎞
S r ,n ( x ) = ∑
⋅ x r ⋅ ⎜1 − ⎟
⎝ c⎠
r = 0 r!(n − r )!
n
n−r
The individual terms of the Bernstein polynomials with increasing order n can be
illustrated by means of a Pascal’s triangle. See Fig.1.
3
(3)
“Partition of Unity”
1
∑ S (x ) = 1
n
r =0
1− x
r ,n
(1 − x) 2
(1 − x)3
(1 − x) 4
(1 − x)5
Leading Edge
Radius
n=0
5(1 − x) 4 x
2(1 − x) x
3(1 − x) 2 x
4(1 − x)3 x
n=1
x
n=2
3(1 − x) x 2
6(1 − x) 2 x 2
10(1 − x)3 x 2
x2
x3
4(1 − x) x 2
10(1 − x) 2 x 3
“Shaping Terms” which do not effect
leading or trailing edge geometry
5(1 − x) x 4
n=3
x4
n=4
x5
n=5
Trailing Edge
Boattail Angle
Figure 1: Bernstein Polynomial Representation of the Unit Shape Function
The first term of equation 3, when written in polynomial form as shown in Figure 1,
defines the leading edge radius and the last term the trailing angle. The other terms are
shaping terms which do not influence airfoil leading or trailing edge shape.
The entire airfoil can be represented by one upper and one lower unit class function
multiplied by a unit shape function with Bernstein polynomials (BP). In Ref.16 it was
found that BP of the order n of 6-9 matched the airfoil geometries and aerodynamic
forces. It was further suggested that for optimization purposes the order n could be
lowered to 4 which reduces the design variables and thus improves computation times.
Based on the findings in Ref.16 the order of the BP is set to n=4 for all further propeller
optimizations.
Equations 3 and 4 define the upper and lower airfoil geometry. The first two terms are
the class function which sets the leading and trailing shape of the airfoil. The remaining
terms are the BP for n=4 with the individual coefficients. See also Figure 1. When thin
airfoil propellers are considered the average of the upper and lower airfoil geometry
functions is taken to define the mean chord line on which a single layer of vortex
elements is placed.
4
Upper surface definition:
N1
N2
x⎞
⎛ x⎞ ⎛
yupper ( x ) = ⎜ ⎟ ⋅ ⎜1 − ⎟ ⋅
⎝c ⎠ ⎝ c ⎠
4
3
x⎞ ⎛ x⎞
x⎞
⎛ x⎞
⎛
⎛
A1 ⋅ ⎜ ⎟ + 4 ⋅ A2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + 6 ⋅ A3 ⋅ ⎜1 − ⎟
⎝ c⎠
⎝c ⎠
⎝ c ⎠ ⎝c ⎠
[
3
2
x⎞ ⎛ x⎞
x⎞
⎛
⎛
+ 4 ⋅ A2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + A5 ⋅ ⎜1 − ⎟
⎝ c ⎠ ⎝c ⎠
⎝ c⎠
4
2
⎛ x⎞
⋅⎜ ⎟
⎝c ⎠
2
(4)
]
Lower Surface definition:
N3
N4
⎛ x⎞ ⎛ x⎞
ylower ( x ) = −⎜ ⎟ ⋅ ⎜1 − ⎟ ⋅
⎝c ⎠ ⎝ c ⎠
[
4
3
2
⎛ x⎞
⎛ x⎞ ⎛ x⎞
⎛ x⎞ ⎛ x⎞
A6 ⋅ ⎜ ⎟ + 4 ⋅ A7 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + 6 ⋅ A8 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟
⎝c ⎠
⎝ c ⎠ ⎝c ⎠
⎝ c ⎠ ⎝c ⎠
3
2
⎛ x⎞ ⎛ x⎞
⎛ x⎞
+ 4 ⋅ A9 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + A10 ⋅ ⎜1 − ⎟
⎝ c ⎠ ⎝c ⎠
⎝ c⎠
4
2
]
with c being the unified chord, x the local variable ranging from 0 – 1 and A1-10 the
coefficients ( with a range of 0 - 1). This 2-D shape function requires 14 variables.
3D Propeller Geometry
The class/shape function methodology of representing 2D airfoils is extended here to
define general 3D propeller shapes. This approach is based on the work done by Ref.15
and 16 and then extended further to allow for a more open design space in the creation of
general propeller geometries. To accomplish spanwise geometry variation the coefficients
A1-10 and N1-4 for the top and the bottom airfoil side must be dependent on the local
spanwise position. For each of the dependent variables A1-10 a BP with n=3 and for N1-4 a
BP with n=1 is introduced, to provide a smooth spanwise coefficient transition. Thus the
variables for the upper and lower airfoil geometry are
5
(5)
3
2
y⎞ ⎛ y⎞
⎛
⎛ y⎞
⎛ y⎞
Au _ i⎜ ⎟ = [ au i ⋅ ⎜ ⎟ + 3 ⋅ au i + 5 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ +
⎝ b ⎠ ⎝b ⎠
⎝b ⎠
⎝b ⎠
2
y⎞
y⎞ ⎛ y⎞
⎛
⎛
3 ⋅ au i +10 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + au i +15 ⋅ ⎜1 − ⎟
⎝ b⎠
⎝ b ⎠ ⎝b ⎠
3
]
3
2
y⎞ ⎛ y⎞
⎛
⎛ y⎞
⎛ y⎞
Al _ i⎜ ⎟ = [ al i ⋅ ⎜ ⎟ + 3 ⋅ al i + 5 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ +
⎝ b ⎠ ⎝b ⎠
⎝b ⎠
⎝b ⎠
2
y⎞
y⎞ ⎛ y⎞
⎛
⎛
3 ⋅ al i +10 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + al i +15 ⋅ ⎜1 − ⎟
⎝ b⎠
⎝ b ⎠ ⎝b ⎠
y⎞
⎛
⎛ y⎞
⎛ y⎞
N j _ u ⎜ ⎟ = nu j ⎜ ⎟ + nu j + 2 ⎜1 − ⎟
⎝ b⎠
⎝b ⎠
⎝b ⎠
y⎞
⎛
⎛ y⎞
⎛ y⎞
N j _ l ⎜ ⎟ = nl j ⎜ ⎟ + nl j + 2 ⎜1 − ⎟
⎝ b⎠
⎝b ⎠
⎝b ⎠
3
(6)
]
b is the unit propeller radius, y the local spanwise station, i and j the variable counters
from 1-5 and 1-2.
The total number of variables to describe the upper and lower airfoil geometry is 44.
Twenty for each, the upper and lower airfoil shape function and 4 variables to define the
two class functions. The unit class/unit shape function for the upper airfoil side is
⎛ x⎞
y upper ( x, y ) = ⎜ ⎟
⎝c ⎠
[
N1 u ( y )
x⎞
⎛
⋅ ⎜1 − ⎟
⎝ c⎠
N2u ( y )
⋅
4
3
2
x⎞ ⎛x⎞
x⎞ ⎛ x⎞
⎛x⎞
⎛
⎛
A1 ( y ) ⋅ ⎜ ⎟ + 4 ⋅ A( y )2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + 6 ⋅ A3 ( y ) ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟
⎝c ⎠
⎝ c ⎠ ⎝c ⎠
⎝ c ⎠ ⎝c ⎠
x⎞
⎛
+ 4 ⋅ A2 ( y ) ⋅ ⎜1 − ⎟
⎝ c⎠
3
2
x⎞
⎛ x⎞
⎛
⋅ ⎜ ⎟ + A5 ( y ) ⋅ ⎜1 − ⎟
⎝c ⎠
⎝ c⎠
4
2
]
An angle of attack (AOA), chord length and sweep variation function is added to the
unit class/shape function to further extend the propeller design space. All three functions
are defined in the same manner as the unit shape function variables A1-5 before, with the
addition of a multiplication factor to define the max value of the function. Each function
requires 5 additional variables, four to define the BP with n=3 and one to set the max
value of the function.
6
(7)
The angle off attack, chord length and sweep function can be written as:
3
2
y⎞ ⎛ y⎞
⎛
⎛ y⎞
AOA( y ) = aoa5 ⋅ [ aoa1 ⋅ ⎜ ⎟ + 3 ⋅ aoa2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ +
⎝ b ⎠ ⎝b ⎠
⎝b ⎠
2
y⎞ ⎛ y⎞
y⎞
⎛
⎛
3 ⋅ aoa3 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + aoa4 ⋅ ⎜1 − ⎟
⎝ b⎠
⎝ b ⎠ ⎝b ⎠
3
]
3
(8)
2
y⎞ ⎛ y⎞
⎛ y⎞
⎛
Chord ( y ) = chord 5 ⋅ [ chord1 ⋅ ⎜ ⎟ + 3 ⋅ chord 2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ +
⎝b ⎠
⎝ b ⎠ ⎝b ⎠
y⎞
⎛
3 ⋅ chord 3 ⋅ ⎜1 − ⎟
⎝ b⎠
2
y⎞
⎛ y⎞
⎛
⋅ ⎜ ⎟ + chord 4 ⋅ ⎜1 − ⎟
⎝b ⎠
⎝ b⎠
3
]
3
2
y⎞ ⎛ y⎞
⎛ y⎞
⎛
Sweep( y ) = sweep5 ⋅ [ sweep1 ⋅ ⎜ ⎟ + 3 ⋅ sweep 2 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ +
⎝b ⎠
⎝ b ⎠ ⎝b ⎠
2
y⎞ ⎛ y⎞
y⎞
⎛
⎛
3 ⋅ sweep3 ⋅ ⎜1 − ⎟ ⋅ ⎜ ⎟ + sweep 4 ⋅ ⎜1 − ⎟
⎝ b ⎠ ⎝b ⎠
⎝ b⎠
3
] ⋅ ⎛⎜⎜ root chord ⎞⎟⎟
⎝
⎠
with aoa5 and chord5 being the max angle of attack and max chord length. The variable
sweep5 is given in multiples of the propeller root chord. This brings the total number of
variables to 66. Sixty three for the blade geometry, one for the propeller radius, one for
number of propeller blades and one for the position, about which the individual spanwise
airfoil sections are rotated.
Figure 2 shows the geometry of a random propeller generated by the described
scheme.
Figure 2: Propeller Geometry of Class/Shape Function Scheme
7
(9)
(10)
Grid Generation
The surface of the propeller is divided into chordwise and spanwise ring panels. A
single ring panel element is constructed from two neighboring spanwise and chordwise
vortex elements. Control points are located in the center of the ring panel1. The control
points are the locations where the influence of the individual vortex elements is
evaluated. The number of panel elements is fixed throughout the optimization and can be
set in the variable definition section of the program. In the spanwise and chordwise
direction the location of the grid points follows Jackson and Fiddes17 which use a half
cosine distribution spanwise and a full cosine panel distribution chordwise. More detailed
description of panel distribution functions can be found in Refs.17 – 18. An investigation
on panel density to find the minimum required number of panels, but which produces
accurate results to within 0.5%, led to 7 chordwise and 14 spanwise panels. This is
confirmed by Ref. 5 which uses 7 panel elements streamwise and 10 elements spanwise.
Ring and Horseshoe Elements
The ring panel elements consist of four straight vortex lines which form a quadrilateral
and are arranged in a geometric closed form. To fulfill Lord Kelvin’s theorem the
circulation along the panel sides is constant. In this paper only thin airfoil propellers are
considered, thus a single layer of vortex ring elements is placed on the mean chamber line
to simulate the propeller. The upper and lower propeller airfoil geometry, especially the
propeller thickness, does not have any influence on the propeller performance besides
describing the mean chord line. The vortex ring elements cover the entire mean chamber
line, excluding the last chordwise panel, which is a horseshoe element. More detail on
that constraint is found in Refs. 19 and 20.
The wake of the propeller is simulated by horseshoe vortices which extend from the
last chordwise panel two revolutions downstream. The pitch of the horseshoe elements is
constant, which coincide with Goldstein’s21 helical vortex model, and follows the trailing
edge panel to satisfy the Kutta condition. Slipstream contraction is neglected to simplify
the geometry of the wake. See Figure 3 below.
8
Nodal point
Blade trailing edge
Quadrilateral element
Γi
Γi+2
Control point
Horseshoe element
Figure 3: Blade discretization into quadrilateral and horseshoe elements
Aerodynamic noise prediction model
The aerodynamic noise prediction of propellers dates back to 1919 when Lynam and
Webb first published their work, which was driven by the requirement to have aircraft
flying undetected over enemy territory. Since then several different methods of propeller
noise prediction have been developed and continuously improved. In recent years
propeller noise has again become of public interest due to the rapid increase in air traffic
and the popularity of wind turbines for electric power generation. A summary of different
methods can be found in Ref. 22.
The noise prediction scheme used in this optimization is based on an empirical model
developed by Smith23. This model predicts the far field noise during a fly over of a small
general aviation single and twin engine airplane. Based on the data obtained from 30
single and 28 twin engine airplane, Smith developed a single equation, using regression
analysis, to predict propeller noise. The claimed 2 sigma confidence was ± 2.2 [dBA] for
the investigated flyover at 1000 feet AGL.
The required inputs are power [hp], Propeller Speed [rpm], number of propeller
blades, number of engines, flight speed, twist of the propeller tip [deg] and blade
thickness to chord ratio at the 95 % radial blade station.
The noise is then predicted by
9
L A = 29.988 + 2.5796 ⋅ (# engines ) − 0.21156 ⋅ (# blades ) + 128.37 ⋅ (log(M th )) +
⎡ prop tip thickness ⎤
19.009 ⋅ (log(hp )) + 141.338 ⋅ ⎢
⎥ + 0.03806 ⋅ ( prop tip twist )
prop chord
⎢⎣
⎥⎦
(11)
with
rpm
720
1.688 ⋅ (KCAS )
Vp = π ⋅ D ⋅
VA =
518.7
σ⋅
OAT + 460
2
M th =
V A + VP
2
49 ⋅ OAT + 460
D is the propeller diameter [inch], KCAS is airplane speed in knots calibrated, σ pressure
ratio p/po at airplane altitude, OAT outside air temperature [ºF], VP propeller rotational
tip speed, VA airplane true airspeed and Mth helical tip mach number.
Genetic Algorithm
Genetic algorithms have become increasingly popular in initial design optimizations
since they are self contained programs, which can be applied to any analysis.
For the optimization, a binary encoded GA based on the tournament method,
developed by Anderson6-8, is chosen to allow for robust optimization of the propeller
parameters. The fundamental building block of the GA is that it behalves like a ‘learning
program”. The GA uses supposition as a basic mechanism for improvement in a given
problem. Applied to the propeller, the GA optimizes performance through random
variation of the 66 design parameters. For this effort, the parameter to be maximized is
thrust output for a given power setting and flight condition and in some of the analysis
aerodynamic noise generated by the propeller, is considered.
Validation of the Propeller Performance Model
The numerical scheme was validated by comparing results from Ref.5 where a three
bladed NACA 109622 propeller was compared to experimental data. In Ref.5 numerical
prediction was done using the lifting line method with and without 2D drag data
included. In this effort the lifting surface method was used to predict the performance of
the NACA109622 straight blade propeller. The blades are modeled by ten spanwise and
two chordwise panels with a blade angle β=45.4° at the 75% propeller radius. The
10
(12)
geometry of the propeller is given in Ref.5. Figure 4 shows the propeller geometry with
the horseshoe trailers extending two full propeller rotations downstream.
Z
X
Y
Figure 4: NACA 109622 propeller geometry with wake
The results are plotted in Figures 5 and 6 and incorporate experimental data as well as
data from the lifting line method from Cheung3 with and without 2D drag data. The
lifting line method with no drag data and the lifting surface method agree very good,
which is due to the fact that the propeller blade is non chambered thus the lifting surface
method does not improve the accuracy of the solution.
Small discrepancies are observed at higher advance ratios, where the lifting surface
solution under predicts the thrust and power coefficient. This is because the lifting line
method cannot describe the propeller tip geometry properly whereas the lifting surface
method with vortex ring element accounts for it.
11
0.2
0.18
0.16
Thrust Coefficient ct [-]
0.14
0.12
0.1
0.08
Experiment
Lifting Line Method no Thickness and cd=0
Lifting Line Method no Thickness with Drag Data
Lifting Surface Method no Thickness and cd=0
0.06
0.04
0.02
0
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
Advance Ratio [-]
Figure 5: Propeller Thrust Coefficient at various Advance Ratios
0.45
0.4
Experiment
0.35
Lifting Line Method no Thickness and cd=0
Lifting Line Method no Thickness with Drag
Data
Lifting Surface Method no Thickness and cd=0
Power Coefficient [-]
0.3
0.25
0.2
0.15
0.1
0.05
0
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
Advance Ratio [-]
Figure 6: Propeller Power Coefficient at various Advance Ratios
12
2.5
Optimization Results
Case 1: Small Diameter Propeller Optimization
This case investigates the propeller performance improvements through aerodynamic
shape optimization by comparing wind tunnel data of a CAM 13x7 propeller to single
and multipoint optimizations. The two bladed propeller was tested over a free stream
velocity range of 0-60 [feet/sec] with a constant power setting of P=0.09 [hp]. The logged
data included the propeller thrust [lb], the torque [inch lb], the free stream velocity
[feet/sec], the power [Watts] and the propeller rotational speed [rpm]. The electric motor
efficiency is obtained by multiplying the torque with the rotational speed divided by the
power input. Multiplying the motor efficiency with the power input, produces the shaft
power for the optimization runs.
In the first optimization the objective function was set to match the shaft power of the
CAM 13x7 propeller of 48 [W] and at the same time maximize the propeller thrust at a
free stream velocity of 60[feet/sec]. The propeller diameter was limited to match the
tested CAM 13x7, the number of blades was fixed to two blades and the rotational speed
was set to 4800 [rpm]. The parameter of interest is the thrust of the propeller. Figure 7.
shows the single point optimized propeller and wake geometry at 60 [feet/sec] free
stream velocity.
Z
Y
X
Figure 7: Single Point Optimization at 60 [feet/sec] free Stream Velocity
The optimization was run over 500 generations with 400 members per generation. The
thrust of the optimized propeller at 60 [feet/sec] and 48 [W] power input was T=0.53
13
[lbt], which is about 300 % better than the tested propeller. This is due to the fact that the
CAM 13x7 is optimized for takeoff and not cruise condition.
In the second optimization an additional operational point at 20 [feet/sec] free stream
velocity of the CAM 13x7 propeller was added to the 60 [feet/sec] condition. This time
the second objective function was set to match 45 [W] power input, the rotational speed
was set to 3600 RPM and thrust was to be maximized at both 20 and 60 [feet/sec] free
stream velocity while the pitch of the propeller stayed constant. Thrust was improved in
both cases to T20=1.02 [lbt] and T60=0.48 [lbt] when compared to the experimental data.
The third optimization run simulated a variable pitch propeller at 4800 rpm with a
power input of 48 [W] at 20 and 60 [feet/sec] free stream velocity. Maximizing thrust
was again the objective function while matching power input. This optimized propeller
was the most overall efficient one when compared to all prior runs. The thrust at 20
[feet/sec] and 60 [feet/sec] was T20=0.516 [lbf] and T60=1.07 [lbf]. See Figure8 below.
1.4
Fixed pitch CAM 13x7
Optimization 1
Optimization 2
Optimization 3
Variable Pitch CAM 13x7
1.2
Thrust [lbf]
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
Free Stream Velocity [feet/sec]
Figure 8: Propeller Thrust Comparison of CAM 13x7 and Optimized Propellers
Case 2: Optimization of small UAV Propeller
In the second case a propeller for a small UAV was optimized for a given cruise speed
of 60 [mph] and a required thrust of Treq=1.6 [lbf] while thrust for the launch condition of
20 [mph] had to be maximized. The propeller rotational speed was set to 12000 [rpm] and
the diameter was limited to 13 [inches]. Three different optimization runs were
investigated. The first one considered a fixed pitch propeller setup, the second a variable
pitch propeller design which used the optimized launch condition thrust obtained from
14
the first run. In the last run a variable pitch propeller design was considered similar to run
2, but it included compressibility following the Prandtl-Glauert similarity rule. The
results of the optimizations are shown in Table 1.
Optimization 1
Vinf= 20 [mph]
Vinf= 60 [mph]
Vinf= 20 [mph]
Vinf= 60 [mph]
Vinf= 20 [mph]
Vinf= 60 [mph]
Optimization 2
Optimization 3
T20=4.04 [lbf]
T60=1.6 [lbf]
T20=4.13 [lbf]
T60=1.6 [lbf]
T20=4.12 [lbf]
T60=1.6 [lbf]
P20=284.4 [W]
P60=199.5 [W]
P20=294.5 [W]
P60=208.2 [W]
P20=234.6 [W]
P60=205.0 [W]
Table 1: Optimization Results small UAV Propeller
The thrust in optimization runs 2 and 3 are matched very close when compared to case
1.When comparing the power required of run one and two it would be expected that the
variable pitch propeller requires less power at the same thrust levels, but the results show
that the variable pitch propeller requires more power. The explanation can be found in
Figure 9, which shows the convergence behavior of the different objective functions
during the optimization. The objective function of the fixed pitch propeller converges
after about 340 generations where as the variable pitch propeller objective function
seemed to show no convergence behavior. Case three which included compressibility,
shows some improvements when compared to case one with respect to power required.
This is also observed by the lower value of the objective function of case 2 when
compared to case 3. A final analysis can only be done after more extended optimization
runs.
1.4
Objective Function [-]
1.3
1.2
1.1
Variable Pitch Propeller
Fixed Pitch Propeller
Variable Pitch Propeller with Compressibility
1
0.9
0.8
0.7
100
200
300
400
Number of Generations [-]
Figure 9: Objective Function Convergence Behavior
15
500
Case 3: Optimization of General Aviation small Propeller
The last case investigates the performance and noise improvement by matching a
Cessna 172 RG general aviation airplane propeller. Three different runs were made to
determine the influence of compressibility and noise on performance when included in
the objective functions. The propeller rotational speed was set to 2700 rpm, the max
diameter was limited to 75.5 inches, the free stream velocity was 156 mph and the power
was set to match 180 hp in the objective functions.
In case one the objective function included matching the power of 180 hp and
maximizing the thrust. The second run included a second objective function to reduce the
propeller noise and at the same time maximize performance while matching the power
input of 180 hp. The third case was set up as case 2 but compressibility effects were
included. The weighting factor between noise and performance was equal. Table 2 shows
the results of the optimization runs and Figure 10-12 displays the optimized propeller
geometries.
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Cessna 172 RG
Optimization 1
Optimization 2
Optimization 3
Thrust [lbf]
396.7
386.4
375.8
Noise [dBA]
73.5
72.3
68.8
68.3
# Propeller Blades
2
2
3
4
Table 2: Optimization Results small General Aviation Aircraft
The optimization case one shows the best thrust results with a two bladed propeller
which agrees with the general propeller theory that propeller with lower blade number
have generally higher efficiencies. See Figure 9 for propeller geometry. The second run
which included the reduction of propeller noise has a reduced thrust but at the same time
decreased the propeller noise by 4.7 dBA when compared to the experimental data
obtained from a C172 RG. The added third blade shows also agreement with common
theory that a higher number of propeller blades have a lower noise level. See Figure 11.
The last optimization run included in addition to thrust and noise optimization also
compressibility effects which can be seen in the reduced thrust values when compared to
case 2. The plot of the objective functions over the number of generations is shown in
Figure 13 and indicates that longer runs need to be done before reaching convergence.
16
Z
Figure 10: Case 1 two Bladed Optimized Propeller
Z
Y
X
Figure 11: Case 2 three Bladed Optimized Propeller
17
Z
Y
X
Figure 12: Case 3 four Bladed Optimized Propeller
0.775
Objective Function [-]
0.77
Optimization run 2
Optimization run 3
0.765
0.76
0.755
0.75
0.745
0
100
200
300
400
Number of Generations [-]
Figure 13: Case 2 and 3 Objective Function Behavior
18
500
Conclusion and Future Work
It has been shown that performance of thin airfoil subsonic propellers can be
accurately and efficiently modeled using the vortex lattice method. Compressibility
effects, when taken into account show a small reduction in propeller performance at
Mach numbers up to 0.7. It has been shown that this prediction scheme is viable as a
prediction tool used in a Genetic Algorithm based optimization due to fast and accurate
computations. The optimization routine which included the aerodynamic propeller
performance and empirical noise analysis is a first step towards a true multidisciplinary
design optimization. Results shown demonstrate the trade off between a quiet and
aerodynamically efficient propeller.
This propeller design approach could be improved by adding a two layered vortex
model to account for thick airfoils, a model to account for flow separation and a model to
account for viscous effects. The empirical noise prediction scheme which is limited to
general small aviation aircraft propellers could be changed to a source noise model to
allow the noise prediction of a wide variety of propellers.
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