designofquietuavpropellers adissertation ...dk871sj7273/thesis-augmented.… ·...

DESIGN OF QUIET UAV PROPELLERS

A DISSERTATIONSUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND

ASTRONAUTICSAND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF ENGINEER

Alex StollJune 2012

This dissertation is online at: http://purl.stanford.edu/dk871sj7273

© 2012 by Alex Morgan Stoll. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

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http://purl.stanford.edu/dk871sj7273

Approved for the department.

Ilan Kroo, Adviser

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this thesis in electronicformat. An original signed hard copy of the signature page is on file in University Archives.

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Abstract

Extensive recent development and deployment of electric-powered unmanned aerialvehicles (UAVs) has placed increased focus on the design of quiet propellers for thesevehicles. To aid in this effort, a design methodology is developed to synthesizelow-noise and efficient propellers for electric UAVs. This methodology employs a blade-element momentum theory performance analysis method, a compact-chord aeroacousticanalysis method based on Farassat’s Formulation 1A of the Ffowcs Williams-Hawkingsequation, and a beam analogy aeroelastic model. Blade geometries that minimizeinduced and profile losses are analytically determined, and other design parametersare chosen parametrically to balance noise reduction and aerodynamic performance.Various noise-reduction techniques and their impacts on propeller performance areanalyzed, and reduced tip speeds and increased blade counts are selected as mostpromising for the chosen conditions.

Two propellers of different blade counts designed using this methodology are manu-factured to validate the methodology, and static test stands are developed to performthis validation. Comparisons between the predicted and experimental performancereveal a deficit in thrust; this is partially explained by the inaccurate wake geometryassumptions of the blade-element momentum theory at static conditions. The remain-der of this discrepancy is likely attributable to a combination of experimental errorand rotational and three-dimensional aerodynamic effects not analytically modeled. Asignificant noise reduction was experimentally demonstrated between propellers of lowand high blade counts, validating trends identified analytically; this reduction, whilelarge, was less than the predicted magnitude, likely due to manufacturing irregularitieslimiting destructive acoustic cancellation between propeller blades. Although the

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performance discrepancy precludes the use of this methodology for the design ofproduction propellers, the methodology is valuable in easily identifying practical quietpropeller configurations for preliminary design studies.

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Acknowledgments

I am immensely grateful to the Stanford UAV Center, which funded this work.I would like to thank my thesis advisor, Professor Ilan Kroo, for his invaluable

guidance and support. It has been a great pleasure working with someone possessingsuch technical insight. I also owe a sizable debt to Professors Juan Alonso and KarthikDuraisamy for altruistically providing me with much-needed advice from their ownexpertise.

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Nomenclature

Roman Symbols

Cf force coefficient

M∞ free-stream Mach number

x acoustic source location

xo acoustic observer location

A blade section cross-sectional area

A(f) A-weighting function

a0 free-stream sonic velocity

c chord

Cd local section drag coefficient

Cl local section lift coefficient

CP power coefficient

CT thrust coefficient

Cmc/4 moment coefficient about the quarter-chord

D section viscous drag

E tensile modulus

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F momentum loss factor

f frequency

fm frequency of harmonic m, where f1 is the rotational frequency

Fcz centrifugal force component in the radial direction

FM figure of merit

G shear modulus

Ix second moment of area with respect to the x-axis

Iy second moment of area with respect to the y-axis

Ixy area product of inertia

J advance ratio (J = πV ∗∞)

K torsion constant

m harmonic

MT blade tip Mach number

Mt local torsion

Mx section moment about the x-axis

My section moment about the y-axis

pl pressure disturbance due to loading noise

pt pressure disturbance due to thickness noise

qcy local normal centrifugal force component

R propeller radius

rh hub radius

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rs tip split location

Re Reynolds number

sy tensile yield strength

t section thickness

V∞ free-stream velocity

Vb local total velocity

VT blade tip speed

Vθi circumferential induced velocity

Vxi axial induced velocity

Greek Symbols

α geometric angle of attack

β geometric blade twist angle

βe elastic twist angle

Γ circulation

γ heat capacity ratio

ν free-stream kinematic viscosity

φ inflow angle

φt inflow angle at the tip

ψ blade azimuth

ψo blade visual azimuth

ρ free-stream density

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ρm density of the blade material

σ normal stress

σc normal stress resulting from centrifugal force

σd design normal stress

τ shear stress

τd design shear stress

θ rake angle

θo acoustic observer azimuth

θs tip split angle

ε section drag-to-lift ratio

Superscripts

′ per unit radius

∗ nondimensionalized (lengths by R, velocities by VT , pressures by ρ)

Subscripts

c centroid location

c/4 quarter-chord location

cg center of gravity location

sc shear center location

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Contents

Abstract iv

Acknowledgments vi

1 Introduction 1

2 Design Goals 42.1 Multidisciplinary Design Considerations . . . . . . . . . . . . . . . . 42.2 Design Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Noise Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Propeller Noise Characteristics . . . . . . . . . . . . . . . . . 62.3.2 Tip Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Blade Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.4 Chord and Twist . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.5 Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.6 Thickness Reduction . . . . . . . . . . . . . . . . . . . . . . . 122.3.7 Unequal Blade Spacing . . . . . . . . . . . . . . . . . . . . . . 132.3.8 Split Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Design Methodology 193.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Design for Maximum Propulsive Efficiency . . . . . . . . . . . . . . . 20

3.2.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Analysis of Arbitrary Designs . . . . . . . . . . . . . . . . . . 27

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3.2.3 2D Section Analysis . . . . . . . . . . . . . . . . . . . . . . . 283.2.4 Split Tip Propellers . . . . . . . . . . . . . . . . . . . . . . . . 283.2.5 Code Description . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.6 Discretization Study . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Aeroacoustic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Code Description . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.3 Discretization Study . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.2 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.3 Code Description . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.4 Discretization Study . . . . . . . . . . . . . . . . . . . . . . . 48

4 Computational Verification Methods 504.1 Analysis Using ANOPP PAS . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Structural Analysis using XROTOR . . . . . . . . . . . . . . . . . . . 51

5 Experimental Validation Methods 565.1 Propeller Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.1 Discussion of Experimental Error . . . . . . . . . . . . . . . . 61

6 Design Examples and Comparisons with Experiment 646.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Propeller Design Specifications . . . . . . . . . . . . . . . . . . . . . . 64

6.2.1 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . 646.2.2 Airfoil Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2.3 Resultant Designs . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Comparison of Acoustic Energy and Total Thrust and Power . . . . . 67

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7 Conclusions 747.1 Summary of Results and Contributions . . . . . . . . . . . . . . . . . 747.2 Consideration of Future Work . . . . . . . . . . . . . . . . . . . . . . 76

A ANOPP Parameters 77

Bibliography 79

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List of Tables

4.1 Comparison with ANOPP PAS integrated loading calculations . . . . 514.2 XROTOR example case . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1 PA 3200 GF material properties . . . . . . . . . . . . . . . . . . . . . 57

6.1 Propeller design specifications . . . . . . . . . . . . . . . . . . . . . . 67

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List of Figures

2.1 Figure of merit and noise sensitivity to MT and B for T = 84 N . . . 72.2 Effect of tip speed and blade count on thickness and loading noise and

figure of merit for 9% thick sections. . . . . . . . . . . . . . . . . . . 82.3 Aeroelastic twist sensitivity to MT and B for T = 84 N . . . . . . . . 82.4 Effect of increasing blade count B on sideline thickness noise due to

destructive interference while holding blade geometry constant . . . . 102.5 Effect of blade sweep on acoustic energy at static conditions for various

blade counts at T = 84 N, MT = 0.36 and R = 0.289 m . . . . . . . . 122.6 Unequal blade spacing: MT = 0.36, R = 0.229 m, CT = 0.22, f1 = 85 Hz 142.7 Unequal blade spacing: MT = 0.5, R = 2.8 m, CT = 0.15, f1 = 9.7 Hz 152.8 Comparison of split-tip (solid) and equivalent conventional (dotted)

platforms (T = 84 N, MT = 0.36, R = 0.289 m) . . . . . . . . . . . . 162.9 Theoretical acoustic impact of split blade tips . . . . . . . . . . . . . 172.10 Split tip noise comparison for B = 5, θs = 30◦, r∗s = 0.75 . . . . . . . 18

3.1 Flow geometry and force coefficients for a blade element . . . . . . . 223.2 Typical results from a series of XFOIL runs at one section . . . . . . 293.3 Error in blade geometry and loading relative to Ne = 67 . . . . . . . 313.4 Example source and sink locations for thickness noise analysis, showing

the division of the section into two panels . . . . . . . . . . . . . . . . 333.5 Observer location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Error in total acoustic energy relative to Ne = 67 . . . . . . . . . . . 383.7 Error in total acoustic energy relative to Nt = 2048 (Ne = 20, No = 51) 393.8 Error in total acoustic energy relative to No = 51 (Ne = 20, Nt = 1024) 40

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3.9 Forces on a blade element . . . . . . . . . . . . . . . . . . . . . . . . 433.10 Force and moment locations for a blade element . . . . . . . . . . . . 453.11 Error in aeroelastic constraints relative to Np = 1024 (Ne = 512) . . . 483.12 Error in aeroelastic constraints relative to Ne = 4096 (Np = 151) . . . 49

4.1 Comparison with ANOPP PAS loading calculations at MT = 0.36 . . 524.2 Comparison with ANOPP PAS acoustic calculations (first two harmon-

ics at 4R and θo = 90◦) . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Directivity comparison at 4R with ANOPP PAS acoustic calculations

(first harmonic, MT = 0.36) . . . . . . . . . . . . . . . . . . . . . . . 544.4 Comparison of structural calculations between the present methodology

and XROTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Tensile yield strength and stiffness of select rapid prototyping materials 575.2 Comparison of intended section geometry with measured manufactured

geometry at r∗ = 0.8 for a Q3 blade . . . . . . . . . . . . . . . . . . . 585.3 Test Stand 1 with the APC LP415512 propeller . . . . . . . . . . . . 595.4 Operation of Test Stand 2 . . . . . . . . . . . . . . . . . . . . . . . . 605.5 Sideline noise results from Test Stands 1 and 2 with the LP415512

propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.6 Comparison of experimental performance results for the LP415512

propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Comparison of the current thickness specification to other propellerdesigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Airfoil section geometries, scaled to 15% thickness . . . . . . . . . . . 676.3 Predicted sensitivity of efficiency and noise of an optimum propeller to

airfoil choice (T = 84 N, MT = 0.36, R = 0.289m, V∞ = 0) . . . . . . 686.4 Propeller geometry (planform view) . . . . . . . . . . . . . . . . . . . 696.5 Specifications and predicted performance of the Q3 (blue) and Q9

(green) propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.6 Predicted OASPL directivity patterns . . . . . . . . . . . . . . . . . . 71

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6.7 Sideline noise at 6R and MT = 0.32 . . . . . . . . . . . . . . . . . . . 726.8 Experimental (solid lines) and analytical (dashed lines) performance

comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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Chapter 1

Introduction

Propellers, once the exclusive method of propulsion for effectively all means of (ar-tificial) air transport, have lost much of their former importance and interest toboth engineers and the general public since the dawn of the jet age in the 1950s.However, recent technological advances have produced an explosion in popularity ofefficient, relatively slow-flying unmanned air vehicles (UAVs), a great number of whichemploy propellers for propulsion. These technological advances include improvementsin electric motors and batteries, and more and more UAVs are being designed withelectric motors in place of traditionally-popular piston engines.

When employed as a means of propulsion, propellers have typically been a majorsource of aircraft noise, and with relatively silent electric motors replacing loudcombustion engines, minimizing propeller noise is now more important than ever whennoise is a major operating constraint. For military UAVs, avoiding detection can beessential and noise is an understandable concern, but as electric UAVs continue torise in popularity for urban civilian applications — in roles ranging from cartographyto traffic monitoring to firefighting to geophysical survey and many more — quietoperation will also be quite important to avoid disturbing day-to-day life.

Propeller design methods have been incrementally developed since the birth ofaviation. As early as 1919, A. Betz [3] presented the conditions for a minimum-energy loss propeller design, although this formulation employs many small-angleapproximations and applies to only lightly-loaded propellers. In 1948 Theodorsen [26]

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CHAPTER 1. INTRODUCTION 2

extended these results to heavily-loaded propellers as well. E. Larrabee [15] presentedan improved formulation of these design equations in 1979, although this formulation,as with Betz’s, ignores many viscous terms, employs small-angle approximations, andis accurate only for light loading. In 1994 Adkins and Liebeck [1] extended Larrabee’smethod to eliminate small-angle approximations and the light loading limits; althoughthis formulation also includes more viscous terms, it still employs Betz’s optimalitycondition which does not fully account for viscosity. In 1943 Haines and Diprose [11]applied the calculus of variations to propeller design, and Kroo [14] uses calculus ofvariations to develop the optimality condition with all relevant viscous terms included.

However, none of these methods consider acoustics or structural objectives. His-torically, quiet propellers have often been designed using a serial method, wherein apropeller is designed for minimum energy loss and subsequently modified to reducenoise. Although such serial methods have been used extensively [22] [25], they natu-rally do not ensure an optimal design and do not easily support additional constraints,such as structural considerations.

Such a multi-objective design problem is commonly solved by employing a relativelyhigh-fidelity performance analysis method in an optimization framework. For example,in 1985 Miller and Sullivan [17] described the application of a conjugate directionsoptimization method to a vortex lattice method and a compact source acousticsmodel, ignoring structural analysis. In 2008 Pagano, Federico, and Barbarino [21]described a propeller blade by parameterizing three radial sections and interpolatingbetween them; a genetic algorithm was then coupled with a physics-based surrogateaerodynamic performance model, a computational structural dynamics code, and anaeroacoustics code. In 2009 Gur and Rosen [10] presented a design method employingmulti-disciplinary optimization with a blade-element method and structural andaeroacoustic analysis codes to design quiet propulsion systems for electric unmannedaerial vehicles.

This work differs from the previously-mentioned methods in important ways: itconcentrates on designing propellers at static conditions, where many low-fidelityperformance analysis methods are less applicable; it makes use of parametric analysesto avoid multidisciplinary optimization, therefore sidestepping the difficulty of choosing

CHAPTER 1. INTRODUCTION 3

an objective function most suitably combining performance and aeroacoustic objectives;and, significantly, it experimentally evaluates propellers designed using the methoddescribed herein to analyze the accuracy of this method. Although a significant amountof accuracy is sacrificed by foregoing high-fidelity performance analysis methods,the resulting speed of obtaining a solution makes this methodology attractive forpreliminary design studies.

Chapter 2

Design Goals

2.1 Multidisciplinary Design Considerations

The aim of this work is to develop a methodology capable of designing practical,quiet, efficient propellers suitable for use on electric UAVs. These three design goalscannot necessarily be met simultaneously and often require trades. Practically alldesign decisions that affect noise also affect propulsive efficiency. Therefore, these twoobjectives must be carefully analyzed to avoid making trades that would result in anundesirable design.

Structural and manufacturing considerations impose important constraints. Thescope and budget of this project dictated the choice of material — a choice that greatlyimpacts structural constraints in the form of material strength, stiffness, and, to alesser extent, density. Additionally, manufacturing considerations limit the minimumdesired blade thickness.

2.2 Design Point

Limits on the cost and complexity of potential experimental analysis methods dictatedthe choice of certain design parameters:

• Static conditions (i.e., the flight velocity V∞ = 0) were chosen to eliminate the

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CHAPTER 2. DESIGN GOALS 5

need for wind tunnels.

• The propeller diameter was limited to 1.5 ft (0.457 m) to reduce propeller andtest stand manufacturing costs.

This diameter is similar to that of the propellers of UAVs such as the InsituAerosonde (1.6 ft) and the Boeing ScanEagle (1.4 ft); therefore, a power of 1.5 kWwas chosen since it is similar to the power output of aircraft of this size (1.3 and 1.4kW, respectively, for the Aerosonde and ScanEagle).

Because the standard definition of propulsive efficiency is always zero at staticconditions, the performance will instead be evaluated with a figure of merit (FM)defined as

FM =√

2π

C3/2T

CP

where the thrust and power coefficients are functions of the tip Mach number MT ,free-stream density ρ, blade radius R, free-stream speed of sound a0, and thrust andpower T and P :

CT = T

4ρR2(MTa0/π)2

CP = P

4ρR2(MTa0/π)3

To reduce the complexity of the analysis, the propeller design will only be consideredat a single operating point. Interactions with aircraft structures, such as acousticreflections and inflow and wake effects, depend greatly on the aircraft layout and arebeyond the scope of this work.

2.3 Noise Reduction Methods

While a variety of methods have been shown to reduce propeller noise, many of theseare not applicable to this design point. The effectiveness of select methods and theirimpact on other design considerations is discussed in the following sections.


Unless explicitly stated otherwise, the noise metric used is the total unweightedacoustic energy on a circle of radius 4R surrounding the propeller. Perceived noiselevels, in the form of A-weighted values, are discussed when perceived noise leveltrends differ from unweighted noise level trends.

2.3.1 Propeller Noise Characteristics

Propeller noise is often deconstructed into components categorized by their physicalsource and classified as harmonic or broadband. Harmonic noise produced by propellersdescribes the noise produced at discrete harmonic tones and results from blade thicknessand loading. Broadband noise occurs across a wide range of frequencies and resultsfrom interactions of turbulence with the blades. [16]

Thickness noise is a product of the geometry of the propeller and its motion relativeto the observer. Essentially, it is caused by the blade volumes periodically forcing theair around them as they rotate. Loading noise results from the aerodynamic forceperiodically imparted on the air by the blades as they rotate. Broadband noise can becaused by a different mechanisms. Inflow turbulence, typically a result of a propelleroperating in the wake of a wing or other aircraft structure but also commonly a resultof clean-air turbulence, is one such source. Turbulence in the boundary layer overthe blade surface can produce noisy fluctuating loading near the blade trailing edges.Similarly, turbulence in the core of the blade tip vortex can interact with the bladetrailing edge and produce noise.

In this work, the propeller is assumed to be operating in the absence of inflowturbulence. The other sources of broadband noise are difficult to predict as a conse-quence of their origins in boundary-later and tip vortex turbulence and are commonlyapproximated with semi-empirical relations.

2.3.2 Tip Speed

Tip speed is, generally, greatly effective at controlling all sources of propeller noise.For example, the overall noise in decibels of conventional propellers operating at lowto medium flight speeds has been observed to vary as approximately 40 times the


tip Mach number. [20] However, a typical propeller designed for high efficiency willlikely be operating at the tip speed that produces the highest efficiency (given relevantconstraints), and any reduction of this tip speed will require more power to producethe same thrust. This trade-off may or may not be significant, depending on otherpropeller design characteristics. However, as can be seen from figures 2.1 and 2.2, atip speed of about 400 ft/s or Mach 0.36 is a quite reasonable compromise for theseoperating conditions for a variety of blade counts and for both thin (9%) and thick(15%) sections. Also evident from these figures is the much greater dependence ofnoise than figure of merit on tip speed for most of these values.

(a) 9% thick sections (b) 15% thick sections

Figure 2.1: Figure of merit and noise sensitivity to MT and B for T = 84 N

The shape of the constant-B curves in figure 2.1 depends greatly on the airfoilchoice and Reynolds number, in that the efficiency at higher tip speeds is limitedby the performance of the airfoil at these lower Reynolds numbers. Although theReynolds number increases with speed, this is overpowered by the decrease in bladechord of the optimal design at this higher speed. For larger propellers, which operateat higher Reynolds numbers, the optimal design tip speed for performance will besignificantly higher.

Increasing the tip speed also raises the frequency of the noise, which can impactthe perceived noise even if the actual noise is constant. A comparison between flat-weighted (dB) and A-weighted (dBA) metrics is shown in figures 2.2a and 2.2b; for


(a) dB (b) dBA (c) FM

Figure 2.2: Effect of tip speed and blade count on thickness and loading noise andfigure of merit for 9% thick sections.

these conditions, the trends for these two metrics are the same.Typical propellers are most efficient at near-transonic tip speeds, and therefore

operate in this regime. A tip speed of Mach 0.36 is unusually slow; a drawback ofthis is that empirical data is not available from which to extrapolate broadband noisepredictions. Because broadband noise is often relatively unimportant [16], it will beignored in this analysis; however, it will be analyzed from the experimental data.

Figure 2.3 illustrates the strong impact of aeroelastic considerations on the selectionof tip speed, blade count, and section thickness.

(a) 9% thick sections (b) 15% thick sections

Figure 2.3: Aeroelastic twist sensitivity to MT and B for T = 84 N


2.3.3 Blade Count

Blade count (B) also can have a significant effect on propeller noise. Althoughpropellers with many blades are typically avoided for structural and Reynolds numberreasons, noise reduction was one of the reasons for the recent adoption of the new8-bladed NP2000 propeller for C-2, C-130, and E-2 aircraft. [2] Chusseau et al. showedthat, for example, increasing blade number from 2 to 5 while scaling the blades tomaintain the same power absorption decreased flyover overall sound pressure levelby about 10 dB. [5] Sensitivities of the noise, efficiency, and aeroelastics of optimalpropeller designs to blade count at the present conditions (T = 84 N, R = 0.229 m,V∞ = 0) are shown in figures 2.1-2.3.

One mechanism for the noise reduction potential of high blade counts is the de-structive interference between the acoustic signals from the different blades. Figure 2.4demonstrates the reduction in sideline thickness noise resulting from altering the bladecount while holding the blade geometry and operating conditions constant. Althoughdestructive interference between blades reduces loading noise as well, blade countaffects the loading noise in other ways besides acoustic interference; therefore thisfigure isolates the acoustic interference effect from other effects.

Propeller solidity is fairly insensitive to blade count. This means that increasingblade count by a factor k will reduce the mean blade chord by approximately the samefactor; at constant thickness-to-chord ratio, the volume per blade will be reduced byk2 and the total blade volume will be reduced by k. This reduction in blade volumesignificantly lowers thickness noise, although such high-aspect ratio blades may notbe structurally practical, and at low speeds thickness noise may be fairly insignificantrelative to loading noise.

Increasing blade count also increases the frequency of the noise, which may increasethe perceived noise level despite a decrease in actual noise level. However, as seen inthe A-weighted graph in figure 2.2b, this is not the case at these operating conditions.

The parity of the blade count is an additional consideration. For pusher propellersbehind a straight structure that passes through the propeller’s axis of rotation, whichis a common configuration on UAVs, even-bladed propellers cut through the wake ofthis structure two blades at a time. In this configuration, odd-bladed propellers avoid


Figure 2.4: Effect of increasing blade count B on sideline thickness noise due todestructive interference while holding blade geometry constant

this effect and are typically quieter, so an odd blade count will be preferred in thisdesign procedure when practical.

2.3.4 Chord and Twist

Altering the chord and twist at fixed total thrust has the potential to affect harmonicnoise in two significant ways: reducing the chord will reduce the thickness noise bylowering the blade volume, and modifying the loading distribution will affect theloading noise. However, because a unique chord and twist distribution that producesthe highest-efficiency propeller exists, these alterations necessarily come at the expenseof efficiency. Furthermore, reducing the chord can result in a blade that is structurallyunsound.


However, these chord and twist modifications have, generally, very limited acousticbenefit. For reasonable designs a reduction of at most around 3 dB can be expected,at some cost in aerodynamic performance. [16] This can usually be accomplished byshifting the loading inwards, which has similar acoustic and aerodynamic performanceeffects as reducing the propeller diameter at constant tip speed. [12] Because of themarginal decrease in noise and potentially significant adverse affect on efficiency,altering the chord and twist distributions will not be pursued in this work.

2.3.5 Sweep

Blade sweep has the effect of dephasing the acoustic signals radiating from differentradial blade stations and is quite effective at reducing noise when section Mach numbersare relatively high, such as in an unducted fan in cruise. At low or zero flight speeds,though, the acoustic benefits are marginal at best [16]. D.B. Hanson [12] showed that,at static conditions, the phase shift of the harmonic m for a sweep angle of θs and anobserver at azimuth θo (where θo = 0 is straight ahead of the propeller) is

φs = mBr∗ tan θs (2.3.1)

where the asterisk superscript indicates nondimensionalization, in this case by theradius R. The dependence on m is not useful since, for the operating conditionsunder consideration, the first harmonic is dominant. However, increasing the bladecount B can increase the effect of sweep. This is illustrated in figure 2.5, which showsthe optimum cubic sweep (i.e., the sweep angle is a cubic function of radius) givenstructural constraints for various blade counts.

It is apparent that, at static conditions, sweep is ineffective at low blade countsbut can provide a benefit with a higher number of blades. However, sweep also affectsthe aeroelastic response of the blade as well as the loading. Additionally, the chosenmethod of aerodynamic performance analysis is unable to account for sweep; therefore,the loading results used to calculate the loading noise in figure 2.5 are indeterminablyinaccurate. Because lowering the tip speed and increasing the blade count alone canproduce practical, efficient propellers with greatly-decreased thickness and loading


Figure 2.5: Effect of blade sweep on acoustic energy at static conditions for variousblade counts at T = 84 N, MT = 0.36 and R = 0.289 m

noise, sweep will not be further analyzed.

2.3.6 Thickness Reduction

Reducing the blade thickness is a simple way to reduce thickness noise. Loading noiseis somewhat affected as well, since an optimally-designed propeller will have differentchord and twist distributions for different airfoil sections. The reduction in thickness istypically detrimental to off-design aerodynamic performance and may result in bladesthat experience too much aeroelastic twist or exceed their design stresses.

However, for low-speed conditions, thickness noise is typically relatively unimpor-tant, and the effect on the loading noise of a optimum propeller of altering thicknessis not usually significant. Figure 2.1 shows that changing the thickness can produce


a noticeable effect on efficiency but a very minor effect on noise for most reasonabledesigns.

2.3.7 Unequal Blade Spacing

Unequal blade spacing has the effect of diminishing the tones at the harmonics of theblade passage frequency and introducing tones at other harmonics of the rotationalfrequency f1 (which is given by f1 = MTa0/2πR). This rarely lowers the unweightednoise (OASPL), but it can reduce the perceived (A-weighted) noise by effectivelyshifting the acoustic energy to lower frequencies. The noise difference depends greatlyon the value of f1.

Figures 2.6a and 2.7a show the effect of changing the blade spacing for four-bladedpropellers, where the blades are positioned at azimuths of ±θ1/2 and π ± θ2/2. (Notethat propellers for which θ1 6= θ2 are not inherently balanced.) Figure 2.6 shows alow-tip speed propeller of the diameter considered for this project (18 inches); figure2.7 shows a large, relatively low tip speed design. It is evident from figures 2.6band 2.7b that the spacings with the lowest A-weighted noise occur when θ1 = θ2. Inthese figures, θ without subscripts refers to spacings in which θ = θ1 = θ2.

Figures 2.6 and 2.7 demonstrate that equal spacing produces the lowest unweightednoise. Figure 2.6b shows that, for the propeller of figure 2.6, unequal spacing providesno advantages in A-weighted noise; however, for the propeller in figure 2.7, a moderatereduction of about 3 dBA is possible at θ = 38.7◦.

The greater effect of the A-weighting at lower frequencies is evident when comparingfigures 2.6d and 2.7d.

2.3.8 Split Tip

Splitting the blade tip reduces the compactness of the noise from each blade, in effectachieving some of the benefit of increasing the blade count, but with lower structuraland Reynolds number penalties. Example geometry of a five-bladed split-tip propeller,along with an equivalent conventional propeller, is shown in figure 2.8.

Hanson [12] derived the influence of sweep on the harmonic thickness and loading


(a) dB (b) dBA

(c) Noise difference between unequal and equalspacings

(d) Tonal component magnitudes

Figure 2.6: Unequal blade spacing: MT = 0.36, R = 0.229 m, CT = 0.22, f1 = 85 Hz

noise emitted by a given blade section. (The phase shift in this noise for a propellerin static conditions is given in equation (2.3.1).) By representing an unswept bladewith a symmetrically split tip as the superposition of two half-size blade tips sweptat opposite angles, these equations can be readily modified to analyze split tip noise.If the tips are swept at the angle θ(r), the strengths of the thickness and loadingharmonics are multiplied by cos

[mBr∗ tan(θ)

]. If the tip split begins at r = rs and


(a) dB (b) dBA

(c) Noise difference between unequal and equalspacings

(d) Tonal component magnitudes

Figure 2.7: Unequal blade spacing: MT = 0.5, R = 2.8 m, CT = 0.15, f1 = 9.7 Hz

splits at the constant angle θs, the sweep angle θ(r) is geometrically related to θs by

tan θ(r) = r − rsr

tan θs

For a hypothetical observer location for which the sound waves from all bladesegments are in phase, the noise reduction at harmonic m by a split tip is therefore


Figure 2.8: Comparison of split-tip (solid) and equivalent conventional (dotted)platforms (T = 84 N, MT = 0.36, R = 0.289 m)

given in decibels by

20 log10

[r∗s +

∫ 1

r∗s

cos(mB tan(θs)(r∗ − r∗s)

)dr∗

]

Although the factor of m allows this geometry to strongly affect higher harmonics,the first harmonic is expected to be dominant. The results of this equation forvarious values of mB, r∗s and θs are shown in figure 2.9. Even a relatively impracticalconfiguration (B = 9, θs = 60◦, r∗s = 0.65) only shows a marginal noise reduction ofless than 4 dB, and these results certainly do not validate a split tip geometry as anattractive alternative to increased blade count.

However, the results will differ for observers at different locations, due to the


Figure 2.9: Theoretical acoustic impact of split blade tips

sensitivity to the relative phase of the sound waves originating from different bladesections to the observer location; additionally, this analysis accounts for neitheracoustic interference from the remaining propeller blades nor aerodynamic effects ofsplitting the blade tips.

If the split tip sections retain the thickness-to-chord ratio of an equivalent unsplitblade, the split tip sections will, in total, occupy roughly half the volume of the unsplittip section, resulting in a reduction in thickness noise that, when loading noise isdominant, would be manifest as a reduction in total noise for observer locations inwhich the thickness and loading noise are of similar phase.

Numerical calculations of the thickness and loading noise for the two propellers offigure 2.8 are shown in figure 2.10. The effect on loading noise is insignificant; themarginal total noise reduction results from the diminished thickness noise largely due


to the the lower blade tip volume.

Figure 2.10: Split tip noise comparison for B = 5, θs = 30◦, r∗s = 0.75

However, a comparison with figure 2.1 shows that increasing blade count is a farmore effective and practical noise reduction strategy.

Chapter 3

Design Methodology

3.1 Overview

A blade-element momentum method is employed to calculate the blade chord andtwist distributions that minimize the required power given the tip speed, radius, bladenumber, advance ratio, section geometry, and desired thrust. This must be combinedwith additional methods to take structural, noise, and manufacturing considerationsinto account.

Because the spanwise loading for an efficient propeller will be relatively insensitiveto differences in chord and twist distributions, the loading noise is typically largelyunaffected by chord and twist distribution choice. The thickness noise can be signifi-cantly altered by changing the chord; however, at the design point of MT = 0.36 andV∞ = 0, thickness noise is low and deviations from highly efficient geometries to reducethickness noise have little effect on the total noise. Changing the twist at fixed chordhas little effect on thickness noise at all. Therefore the chord and twist distributionsthat result in the most aerodynamically efficient propeller design will be adopted sincethey will not impart significant acoustic compromises. Noise reduction will instead berealized by varying tip speed and blade number. A maximum local solidity constraintwill be applied to avoid geometries in which the aerodynamic theory is inaccurate.

Structural considerations – namely allowed induced aeroelastic twist and shearand normal stress – are treated as constraints. Centrifugal stiffening can be effected

19

CHAPTER 3. DESIGN METHODOLOGY 20

by introducing rake (dihedral), greatly reducing internal stress without much effect onperformance and acoustics.

Therefore, the design process comprises selecting a tip speed, blade number, andsection geometry that minimize noise without a significant loss of efficiency whilemeeting structural and manufacturing constraints. The optimal chord and twist arethen calculated given these parameters, and the optimum rake angle for centrifugalstiffening is applied.

Details of these design and analysis methods are provided in the following sections.The design codes, in MATLAB format, are published at http://www.AlexStoll.com/

propellers.

3.2 Design for Maximum Propulsive Efficiency

3.2.1 Basic Theory

Several performance analysis methods are commonly employed in propeller design.The following paragraphs discuss, in brief, the methods most appropriate to the presentdesign problem in approximate order of increasing fidelity.

• Blade-Element Momentum (BEM) theory is commonly used in propeller designand analysis due to its relative simplicity and computational efficiency, in manycases without a large sacrifice in fidelity. Three-dimensional aerodynamic effectsare neglected and the blade is treated as a union of radial blade elements(strips). Vortex theory assuming a rigid constant-pitch wake, ignoring wakecontraction and roll-up, allows prediction of induced velocity vectors at each ofthese blade elements. Together with the rotational and free-stream velocities,the aerodynamic forces at each station can then be predicted from section data.A momentum loss function approximates tip losses.

A significant advantage of this method is the ability to analytically derivethe propeller geometry that results in maximum efficiency, obviating the needfor an optimization framework unless other objectives must be simultaneouslyevaluated. However, BEM carries several drawbacks. While quite accurate at

http://www.AlexStoll.com/propellers

http://www.AlexStoll.com/propellers


low to moderate loading at conventional flight speeds (i.e., advance ratios greaterthan around 0.5), it is less reliable at other operating conditions (including,notably, static conditions). Typical formulations do not capture any effectsof blade sweep, rake, and more unusual geometries (for example, particularlyhigh-solidity propellers). To avoid designing a propeller with a solidity higherthan BEM theory applies to, the solidity must be artificially limited.

• Vortex methods compose a special case of the Euler equations, assuming anincompressible potential flow with the wake vorticity confined to a finite numberof nominally helical vortex elements. (Compressibility corrections are oftenincluded.) Wake contraction and roll-up can be modeled, and viscous dragis often approximated by 2D methods. Vortex methods can be classified intoprescribed and free wake methods. Prescribed wake methods, employing an apriori specification of the position of the vortex elements from experiments, aresimple and computationally efficient but are limited by the availability of relevantexperimental data. Because relevant experimental data was not available for thepresent conditions, prescribed wake methods are not practical for this analysis.Although free vortex wake methods require a large amount of computationaltime — about two orders of magnitude higher than BEM methods — they do notrequire an a priori specification of the vortex element positions. However, theydo require various wake parameters (for example, the vortex strength and roll-updistance) to be set by statistical methods, and the lack of relevant experimentaldata precludes the use of free wake methods as well.

• Euler and Navier-Stokes methods are the highest-fidelity analysis methods con-sidered; these methods can model wake contraction and roll-up, compressibilityeffects, three-dimensional aerodynamic effects, and, with Navier-Stokes meth-ods, three-dimensional boundary layers. However, a solution can easily requireover four orders of magnitude more computational time than a BEM solution.Generation of the computational mesh for a propeller at static conditions, whenthe wake and tip vortices are in close proximity to the blades, is a complexand time-consuming endeavor, particularly if the code employs a structured


mesh; unstructured meshes, while potentially far simpler to create, generallyrequire a further order of magnitude of computational time. Additionally, thereis no convenient method to employ these CFD methods to produce an optimumdesign; instead, the analysis must be coupled with some sort of optimizationframework, further increasing the computational expense.

Therefore, a blade-element momentum method was chosen to prescribe a propellergeometry with optimal propulsive efficiency. Although this choice abandons hopesof obtaining very accurate performance predictions, it facilitates a quick and robustdetermination of the propeller configuration. The method employed is adapted fromthe method proposed by I. Kroo [14], which differs from many other popular methodsby including viscous drag in the calculation of optimal loading.

Figure 3.1: Flow geometry and force coefficients for a blade element


Momentum Equations

Figure 3.1 denotes the relevant section flow and force geometry. From this figure,

tanφ = V ∗∞ + V ∗xir∗ − V ∗θi

(3.2.1)

V ∗b = (V ∗∞ + V ∗xi)/ sinφ (3.2.2)

V ∗xi = cosφ(r∗ sinφ− V ∗∞ cosφ) (3.2.3)

where φ is the local inflow angle and Vxi and Vθi are the axial and circumferentialinduced velocity components. The asterisk superscripts indicate nondimensionalizationby the radius R, the tip speed VT , and/or the free-stream fluid density ρ.

The viscous drag per unit radius is given by

D∗′ = dD∗dr∗ = 1

2V∗b

2c∗εCl (3.2.4)

where ε ≡ Cd/Cl.With equations (3.2.2) and (3.2.4), momentum theory analysis, in which the thrust

and power are determined by calculating the change in momentum of fluid elementstraveling in thin, annular stream tubes through the propeller disc, specifies the thrustand power as:

CT′ = π2(V ∗∞ + V ∗xi)

[πFr∗V ∗xi − εBV ∗b c∗Cl/8

](3.2.5a)

CP′ = π3r∗(V ∗∞ + V ∗xi)

[πFr∗V ∗θi + εBV ∗b c

∗Cl/(8 tanφ)]

(3.2.5b)

where F is the momentum loss factor accounting for the loss of lift near the blade tipsthat results from induced effects associated with a finite number of blades.


Circulation Equations

Including the viscous drag given by (3.2.4), the thrust and power per unit radius canalso be expressed as

CT′ = π2

4 B(Γ∗(r∗ − V ∗θi)− V ∗b

2c∗εCl sinφ/2)

(3.2.6a)

CP′ = π3

4 Br∗(Γ∗(V ∗∞ + V ∗xi) + V ∗b

2c∗εCl cosφ/2)

(3.2.6b)

By the Kutta-Joukowski theorem,

Γ∗ = 12V∗b c∗Cl (3.2.7)

With equations (3.2.7) and (3.2.1), equations (3.2.5) and (3.2.6) can be written as

CT′ = π2(V ∗∞ + V ∗xi)[πFr∗V ∗xi − εBΓ∗/4] (3.2.8a)

CP′ = π3r∗(V ∗∞ + V ∗xi)[πFr∗V ∗θi + εBΓ∗/(4 tanφ)] (3.2.8b)

CT′ = π2

4 V∗b BΓ∗(cosφ− ε sinφ) (3.2.9a)

CP′ = π3

4 V∗b BΓ∗r∗(sinφ+ ε cosφ) (3.2.9b)

Combining equations (3.2.8a), (3.2.9a), and (3.2.2) gives the circulation distribu-tion:

Γ∗ = 4πFr∗V ∗xi tanφ/B (3.2.10)

Conditions for Minimum Energy Loss

Minimum torque with fixed thrust can be shown to be achieved when

tanφ = C − εr∗

εC + r∗(3.2.11)


where C is a constant independent of radius.If viscous drag is neglected in equation (3.2.11), the relation becomes the familiar

minimum energy loss condition first stated by A. Betz [3] as early as 1919 (andcommonly used today, e.g. in the procedure put forth by C. Adkins and R. Liebeck [1]in 1994) that, neglecting the contraction of the wake, the vortex sheet is a regularscrew surface (i.e., r tanφ is a constant independent of the radius).

Now the thrust and power coefficients per unit radius can be written as

CT′ = π3(V ∗∞ + V ∗xi)Fr∗V ∗xi(1− ε tanφ) (3.2.12a)

CP′ = π4(V ∗∞ + V ∗xi)Fr∗

2V ∗xi(ε+ tanφ) (3.2.12b)

From these two equations, it follows that

C = CPπCT

Blade Geometry

Equating equations (3.2.7) and (3.2.10) produces the loading distribution

V ∗b c∗ = 8πFr∗V ∗xi tanφ/(ClB) (3.2.13)

If the value of φ is known, then (by examining equation (3.2.3)) the above equationat a given station is a function of only the local lift coefficient. Since equation (3.2.13)plus a choice of Cl will determine the Reynolds number, and the local Mach number isknown from evaluating Vb from equation (3.2.2), this choice of Cl will also determineε from section data. Explicitly,

ReCl = Re08πFr∗V ∗xi tanφ/B (3.2.14)

whereRe0 ≡

Re

c∗V ∗b= a0MTR

ν


and ν is the free-stream kinematic viscosity. Choosing Cl to minimize ε will in mostcases minimize viscous as well as momentum losses, resulting in the highest possibleaerodynamic efficiency. With α known from section data, the twist can be calculatedas β = α+φ. Note that, when F is zero at the tip (such as in the Prandtl momentumloss function), the optimum geometry will always have a tip chord of zero.

Design Procedure

The Prandl momentum loss function

F = (2/π) arccos(e−f ) (3.2.15)

is employed, wheref = (B/2)(1− r∗)/ sinφt (3.2.16)

and φt is the inflow angle at the tip.Once a desired thrust or power is specified, the design proceeds in the following

steps:

1. Estimate an initial value for C. A good starting point is to find the value thatsatisfies

FM + η = CTJ

CP+√

2π

C3/2T

CP= 1

where η = CTJ/CP is the propulsive efficiency and J = πV ∗∞ is the advanceratio. If the propeller is being designed to operate at a given thrust, then thisinitial value of C is given by C = V ∗∞ +

√2CT/π3.

Additionally, estimate initial values of ε for each station.

2. Determine values for φ and F at each blade station by equations (3.2.11) and(3.2.15).

3. Determine V ∗xi from equation (3.2.3).

4. Determine the product ReCl from equation (3.2.14).


5. Determine Mb from Eq. (3.2.2) via Mb = V ∗b MT .

6. Using a section analysis code or a lookup table, find the values of Cl and α thatminimize ε for the calculated values of ReCl and Mb.

7. Find the value of C such that, after recalculating φ, V ∗xi, F , and CT ′, numericallyintegrating CT ′ produces the desired value (CT if the propeller is being designedfor a specific thrust, or CP/πC if the propeller is being designed for a specificpower).

8. If this new value of C is not sufficiently close to the old one and the new valuesof Cl are not sufficiently close to the previous ones (e.g., within 0.1%), startover at step 2 using the new values. A relaxation factor may be employed toexpedite convergence, but if the optimum value of ε can be precisely determined(for example, when using a lookup table), convergence is typically rapid andrelaxation is unnecessary.

9. Compute the chord from c∗ = ReCl/Re0Clr∗, the blade twist from β = α + φ,

and CP or CT from Eq. (3.2.12b) or (3.2.12a).

3.2.2 Analysis of Arbitrary Designs

If the chord and twist distributions are known, the inflow angle can be found an-alytically. Defining φ∞ = arctan(V ∗∞/r∗) and using equations (3.2.2) and (3.2.3),equation (3.2.13) can be written as

BClc∗ = 8πFr∗ tan(φ− φ∞) sinφ (3.2.17)

Because Cl is a function of α and Vb, both of which are functions of φ, and F is afunction of φ, this equation can easily be solved numerically to determine φ. Once φis known, the previously given relations can be used to determine CT , CP , etc.


3.2.3 2D Section Analysis

The 2D viscous panel code XFOIL is employed to calculate the section properties Cdand Cmc/4 (the moment coefficient about the quarter-chord) and angle of attack forprovided section geometry, Cl, Mach number, and Reynolds number. XFOIL employsa higher-order panel method with a linearly-varying vorticity distribution for potentialflow analysis coupled with the integral method, which determines boundary layerparameters by solving a set of differential equations. [6]

3.2.4 Split Tip Propellers

To produce split tip propeller designs, the preceding method is lightly modified. Theblade count B is taken to be a function of radial position such that B(r) at the splitsection is twice the nominal value. This directly affects the calculation of F , ReCl,and Re.

By relying on relations devised for conventional propellers, this method neglectscomplex aerodynamic effects potentially present in a physical split tip blade. However,since these complex interactions would be unlikely to drastically affect the optimalgeometry and loading distribution, the aeroacoustic effects of this configuration canstill be reliably investigated, even if the performance results are suspect.

3.2.5 Code Description

Propeller Design

After initial values of C, Cl, and ε are estimated, a while loop is entered which doesnot break until the convergence criteria are satisfied. The first step in this loop is tocalculate φ, followed by V ∗xi and Mb, and finally ReCl.

The next step is to employ XFOIL to determine the Cl that minimizes ε at eachstation. To accomplish this, XFOIL is run at a linear series of 16 Cl values (and theircorresponding Reynolds number values) at each station centered on the value of Clcalculated in the previous iteration. The span of this linear series is initialized at 0.30and is multiplied by 2/3 each iteration of the while loop (with a lower limit of 0.002)


to narrow in on the optimum value of Cl as the iteration progresses. Typical resultsat one section are shown in figure 3.2, where gaps indicate points where XFOIL didnot converge.

Figure 3.2: Typical results from a series of XFOIL runs at one section

To improve convergence in XFOIL, XFOIL’s routine to add points to the sectiongeometry definition at corners exceeding its angle tolerance ("CADD") is employed,and the geometry is subsequently re-paneled in XFOIL using 160 panels.

To restrain the code from designing propellers with solidities too high for BEM toproduce accurate results, a maximum allowable local solidity is defined as a piecewiselinear function from 1.0 at r∗ = 0.25 to 0.5 at r∗ ≥ 0.75. A minimum possible Clvalue is calculated at each station based on this maximum allowable solidity. If theCl range that XFOIL is to analyze extends below this minimum, the range is shiftedsuch that its minimum is located at the minimum allowed Cl.


If XFOIL does not converge at any of the 16 Cl values in the range, the sectionproperties are interpolated from the sections that did converge. (If the tip has a chordof zero, its properties are always interpolated in this same manner.) In addition to ε,values of α and Cmc/4 calculated by XFOIL for each station are recorded for futureuse.

If the value of ε at a given station has converged to within 0.1% between successiveiterations, the section properties at this station are frozen for the remainder of theprogram, bypassing further XFOIL runs at this station to improve runtime.

Once the optimum values of ε are determined, the value of C is determined byrunning MATLAB’s FMINSEARCH optimization function with the objective

|CT,desired − CT,calculated(C)|2

where calculation of CT,calculated entails the calculation of φ, V ∗xi, F , and CT ′ and thenumerical integration of CT ′. To aid convergence, the value of C used for the successiveiteration is the mean of this value and the value from the previous iteration. Thewhile loop is terminated once both C and the norm of the vector of Cl values haveconverged to within 0.1% between successive iterations.

Finally, values of φ, V ∗xi, F , Re, and V ∗b corresponding to the final value of C aredetermined, allowing calculation of the twist and chord.

Analysis of Arbitrary Designs

If the chord and twist distributions are known, the performance can be calculated bysolving equation (3.2.17). This is done using MATLAB’s FZERO function. To reduceruntimes, a model of Cl(α, r∗) is created by running XFOIL on a series of angles ofattacks at each station at an estimated Re and Mb (making the assumption that Cl(α)at each station is relatively insensitive to the induced velocity), and equation (3.2.17)is solved by interpolating over this model.


3.2.6 Discretization Study

Number of Blade Elements

The locations of the stations defining the blade elements are defined using a cosinedistribution to bias the number of stations near the tip, where the gradients are larger.Each station location ri is given by:

r∗i = r∗h + (1− r∗h) cos(π

2Ne + 1− i

Ne

), i = 1, ..., Ne + 1 (3.2.18)

where rh is the radius of the hub.Figure 3.3 shows the difference in the results of the blade design routine for various

values of Ne. It is apparent that discretizing the blade into more than about 20 bladeelements does not significantly improve the accuracy of the result.

Figure 3.3: Error in blade geometry and loading relative to Ne = 67


3.3 Aeroacoustic Analysis

3.3.1 Basic Theory

As mentioned in section 2.3.1, propeller noise contains both broadband and harmoniccomponents. The Ffowcs Williams-Hawkings equation is a popular and reliable methodfor analyzing the harmonic noise. Farassat’s Formulation 1A [9] of this equation isused to numerically model the noise of subsonic source regions of propellers and rotorsin the time domain in many acoustic codes such as the Aircraft Noise PredictionProgram (ANOPP) [29], Advanced Subsonic and Supersonic Propeller Induced Noise(ASSPIN), and WOPWOP. This formulation is also used in the present analysis andis described below.

Broadband noise results from turbulence and is usually impractical to modeltheoretically; instead, many broadband noise codes (including ANOPP PAS) relyon empirical models. These empirical models are usually derived from propellers inconventional operating conditions (such as moderate tip speeds) and there is littlebasis to extrapolate these models to the lower tip speeds considered here. However,broadband noise is expected to relatively insignificant at these conditions and will beignored in this analysis. This assumption of low broadband noise will then be verifiedexperimentally.

The Ffowcs Williams-Hawkings equation models the acoustic sources as distribu-tions of monopoles, dipoles, and quadrupoles. These source types correspond to linearthickness, linear loading, and nonlinear sources, respectively. Because the nonlinearsources are only important at transonic blade section speeds (namely MT > .85), theywill be ignored in this analysis.

In the thickness noise calculations, the blade segments are modeled as two chordwisepanels, the leading panel comprising “sources” and the trailing panel comprising“sinks.” [24] Employing only one source and sink per segment reduces the complexityof the analysis with only a marginal reduction in fidelity. The strength of this sourceis equal to the mass flux of the fluid displaced by the moving blade segment and, forthis single source, is equal to Mna0ρ, where Mn is the speed (Mach number) of theblade surface in the normal direction. The associated sink represents the mass flux of


the fluid affected by the rear of the blade segment and has a strength −Mna0ρ. Mn isapproximated by

Mn = Mbt

c

where t is the maximum section thickness. The source and sink are offset by a distanceof one half chord and are placed along the chord line at respective distances of c/8and 5c/8 from the leading edge to model the thickness distribution of a typical airfoilsection (see figure 3.4).

Figure 3.4: Example source and sink locations for thickness noise analysis, showingthe division of the section into two panels

The loading noise is more straightforward to model. To reduce complexity, eachblade is approximated as a line source (a chordwise compactness assumption) ofstrength V ∗b 2Cf/2 placed along the quarter-chord, where the blade load coefficientvector Cf is the coefficient of the force of the blade on the fluid (nondimensionalizedby ρV 2

b c∆r/2). This approximation is valid when the wavelength of the generatedsound greatly exceeds the chord (equivalently, when BMT c

∗/2π � 1).An acoustic source at location x on a blade segment at azimuth ψ (the “emission

azimuth”) will produce pressure disturbances due to thickness and loading pt andpl, respectively, which will reach an observer at location xo fixed to the undisturbedmedium (i.e., x∗o = V∞) when the blade is at azimuth ψo (the “visual azimuth”). Thelocations are specified in an observer-fixed (non-rotating) reference frame, and the


two azimuths are related by

ψo(x∗, ψ)− ψ =√[M∞ · (x∗o − x∗(ψ))]2 + (1−M2

∞)|x∗o − x∗(ψ)|2 −M∞ · (x∗o − x∗(ψ))(1−M∞)2/MT

which describes the length of time required for a sound wave from the source to reachthe observer in a moving fluid. In static conditions (M∞ = 0) this simplifies to themore intuitive relation ψo(x∗, ψ) − ψ = MT |x∗o − x∗(ψ)|. Note that sources along asingle blade emitted at a common azimuth of ψ will, in general, reach the observer atdifferent times (different values of ψo).

A nondimensionalized formulation of Formulation 1A for thickness noise ∆pt andloading noise ∆pl for an untapered blade segment of chord c∗ and span ∆r∗ at azimuthψ is

∆pt(x, ψ)p∞

= γ

4π

MTMn

d∗(1−Md)2 +MnMTd

∗M · d +Md −M2

d∗2(1−Md)3

ψ

c∗∆r∗ (3.3.1)

∆pl(x, ψ)p∞

= γ

4πM2

b

2

MT Cf · dd∗(1−Md)2 + Cf · (d−M)

d∗2(1−Md)2 +

(Cf · d)MTd∗M · d +Md −M2

d∗2(1−Md)3

ψ

c∗∆r∗ (3.3.2)

where the subscript ψ indicates that the terms are evaluated at the emission azimuthψ. The dotted quantities indicate derivatives with respect to emission azimuth (e.g.M = ∂M/∂ψ) and

d∗(x∗, φ) = x∗o − x∗(ψ)−V∗∞[ψo(x∗, ψ)− ψ]

d = d/d


M = (x∗ −V∗∞)MT

Md = M · d

Because the thickness noise results from a source and a sink, equation (3.3.1) mustbe evaluated twice, resulting in ∆pt+(xt+, ψ) and ∆pt−(xt−, ψ) for the source andthe sink, respectively, where xt+ and xt− respectively indicate the source and sinklocations.

If the emission azimuths for the thickness source, thickness sink, and loading sourceat blade segment j corresponding to a given visual azimuth ψo are ψt+,j, ψt−,j, andψl,j, respectively, then the total harmonic noise at this visual azimuth is

p(ψo) =∑j

[∆pt+(xt+,j, ψt+,j) + ∆pt−(xt−,j, ψt−,j) + ∆pl(xl,j, ψl,j)

](3.3.3)

The observer pressure-time history for a series of visual azimuths ψo,i can be approxi-mated by determining the visual azimuths associated with a given series of emissionazimuths for each source on blade segment j and using these values to interpolate∆pt+, ∆pt−, and ∆pl for each blade segment onto the values of ψo,i. Although inexact,this approach is much simpler than explicitly calculating the ∆p values at the emissionazimuths corresponding to common visual azimuths.

For rotationally symmetric propellers in symmetric operating conditions (zeroangle of attack and sideslip and uniform inflow), the noise can be first calculated foronly a single blade; the final pressure time history is then a sum of this calculatedpressure, phase shifted the appropriate amount for each blade of the propeller.

Spectral analysis on the resulting pressure-time history is accomplished via a fastFourier transform (FFT) algorithm:

Y = FFT (p(ψo))/Nt

where Nt is the number of points in the discrete sound wave. The RMS Fouriercoefficients Ck are then given by

Ck =√

2|Yk| for k = 1, ..., Nt/2


(The DC component C0 is irrelevant and is neglected.) Sound pressure levels (SPL),A-weighted sound pressure levels (SPLA), and overall sound pressure levels (OASPL)are given in decibels by

SPLk = 20 log10 Ck

SPLA,k = SPLk + A(fk)

OASPL = 10 log10∑k

10SPLk/10

The frequencies fk are given by fk = kMTa0/(2πR) when p(ψo) spans one completerevolution. A(f) is the A-weighting function, given in decibels.

The acoustic energy on a circle centered on the propeller in decibels is

10 log10 2∫ π

0

∑k

10SPLk(θo)/10dθo

where θo is the observer azimuth (shown in figure 3.5.)


First, the source locations x for a single blade are calculated: there are three sourcesper segment, and the locations must be known for the range of emission azimuths ofinterest (typically one complete revolution). At this point the values of M, Mn, and Mare calculated as well. (M is simply MT x.) Mn is found by numerically differentiatingMn:

M i+1/2n = (M i+1

n −M in)/∆ψ

The values of M i+1/2n are then interpolated to find M i

n. Using the same differenti-ation method, Cf is calculated. Next, for each observer location, emission azimuth,and source, ψo, d∗, Md, and ∆p/p∞ are calculated.

An evenly-spaced ψo vector is then defined onto which the ∆p vectors are interpo-lated. To avoid the problem of extrapolating values past the beginning or end of the∆p vectors, the evenly-spaced ψo vector is defined to span two revolutions, and tworevolutions of ∆p are used for the extrapolation (the values in the second revolutionbeing simply repeated values from the first). The middle of the result — i.e., from


Figure 3.5: Observer location

ψ = π to ψ = 3π — is then taken and shifted by π to produce the desired valuesranging from ψ = 0 to ψ = 2π.

Now that the values of ∆p are known at common values of ψo, p(ψo) can be foundby equation (3.3.3). The DC component of p(ψo), given by the mean of p(ψo) across asingle revolution, is removed. The resultant pressure time history for the single bladeis then phase-shifted by 2π(n − 1)/B for each of the blades n = 2, ..., B; the totalsound wave from all the blades is then a sum of the original signal and these shiftedsignals, interpolated onto the original time points.




Based on the results shown on figure 3.6, the blade is divided into 20 elements (21stations) for aeroacoustic calculations, using the same element locations given inequation (3.2.18).

Figure 3.6: Error in total acoustic energy relative to Ne = 67

Number of Waveform Time Plots

Based on the results shown on figure 3.7, the blade revolution is temporally discretizedinto 1024 points for the aeroacoustic calculations.

Number of Observers

The total acoustic energy is calculated by evenly placing observers along a semicircleencircling the propeller and numerically integrating the acoustic energy using the


Figure 3.7: Error in total acoustic energy relative to Nt = 2048 (Ne = 20, No = 51)

trapezoidal rule. Figure 3.8 shows that using more than 11 observers effectively doesnot improve the accuracy; therefore, 11 observers will be used in these calculations.

3.4 Structural Analysis

3.4.1 Introduction

Structural analysis is a vital element of designing a practical propeller. In this work,the structural analysis is performed to evaluate constraints on aeroelastic twist andnormal and shear stresses. The selected propeller manufacturing process described insection 5.1 results in solid blades of uniform isotropic material properties; productionpropeller blades typically employ more complex but efficient constructions whichrequire fundamentally different analysis methods.

Approximating the blade sections as polygons and the blades as slender beamsallows the blade geometric properties, aeroelastic deformations, and internal stressesto be readily estimated. For production propeller designs, a higher-fidelity method,


Figure 3.8: Error in total acoustic energy relative to No = 51 (Ne = 20, Nt = 1024)

such as finite element analysis, should be employed instead, to allow less conservativeconstraints that will provide greater design flexibility. Typical production propellerstructural analysis also entails evaluation of the weight and dynamic structural responseof the propeller; in this work, weight is not an issue and conservative constraintson calculated deflections greatly simplify the problem by eliminating the need fordynamic structural response analysis.

The specific structural constraints imposed are:

• The aeroelastic twist βe will not exceed ±1◦ (max |βe| ≤ 1◦)

• The normal stress will not exceed the design normal stress (max |σ| ≤ σd)

• The shear stress will not exceed the design shear stress (max |τ | ≤ τd)

3.4.2 Basic Theory

A coordinate system is defined for each section such that, if the section were untwistedand at zero angle of attack, x would be oriented in the direction of local drag and


y would be oriented in the direction of local lift. If the blade sections are treated aspolygons and xi and yi are coordinates centered at an arbitrary location (e.g., theleading edge) tracing the section outline counterclockwise such that x0 = xn andy0 = yn, then the cross-sectional area and centroid location can be calculated by:

ai = xiyi+1 − xi+1yi

A = 12

n−1∑i=0

ai

xc = 16A

n−1∑i=0

(xi + xi+1)ai

yc = 16A

n−1∑i=0

(yi + yi+1)ai

Blade section inertial properties about the centroid can then be determined:

x′i = xi − xc

y′i = yi − yc

a′i = x′iy′i+1 − x′i+1y

′i

Ix = 112

n−1∑i=0

(y′2i + y′iy′i+1 + y′2i+1)a′i

Iy = 112

n−1∑i=0

(x′2i + x′ix′i+1 + x′2i+1)a′i

Ixy = 124

n−1∑i=0

(x′iy′i+1 + 2x′iy′i + x′i+1y′i)a′i

Ix is the second moment of area with respect to the x-axis; Iy is the second momentof area with respect to the y-axis; Ixy is the area product of inertia.

If the camber line of a given section is of length U and is broken up into segments


of length dU and thickness t, the torsion constant K for a solid homogeneous sectioncan be approximated by [28]

K = F

3 + 4F/AU2

whereF =

∫ U

0t3dU (3.4.1)

Each section is assumed to be under only aerodynamic and centrifugal loads. Thecentrifugal force at each section acts in the radial (z) direction and is given by

Fcz(r) = ρmV2T

R∫r

A∗(ρ)ρdρ

where ρm is the density of the blade material. The centrifugal force component thatopposes the aerodynamic force in the y direction is

qcy = ρmV2TA∗r sin θ

where θ is the rake (dihedral) angle shown in 3.9.Then the distributed loads are

qx = qfx = 12ρV

2b cCfx (3.4.2a)

qy = qfy − qcy = 12ρV

2b cCfy − qcy (3.4.2b)

where Cfx and Cfy are the coefficients of the aerodynamic force components in thelocal coordinate system.

The bending moments about the x and y axes are calculated by Euler-Bernoulli


Figure 3.9: Forces on a blade element

beam theory:

Mx(r) = −R∫r

Sx(ρ)dρ (3.4.3a)

My(r) = −R∫r

Sy(ρ)dρ (3.4.3b)

where Sx and Sy are shear forces given by

Sx(r) = −R∫r

qy(ρ)dρ (3.4.4a)

Sy(r) = −R∫r

qx(ρ)dρ (3.4.4b)

The normal stress due to bending at point i on the section is given by [23]

σn,i = −(MyIx +MxIxy)xi + (MxIy +MyIxy)yiIxIy − I2

xy


The centrifugal normal stress is

σc = Fcz/A

Then the maximum normal stress due to bending in the section is given by

σz = maxi

(σc − σn,i)

= maxi

σc + (MyIx +MxIxy)xi − (MxIy +MyIxy)yiIxIy − I2

xy

The applied torque at radius r is

Mt(r) =R∫r

Mt0(ρ)[Cmf,x(r, ρ) + Cmf,y

(r, ρ) + Cmc/4(ρ) + Cmc(r, ρ)]dρ (3.4.5)

whereMt0 = 1

2ρV2b c

2

and

Cmf,x(r, ρ) = Cf,x(ρ)−[ysc(r)− yc/4(ρ)]

c(ρ) (3.4.6)

Cmf,y(r, ρ) = Cf,y(ρ)xsc(r)− xc/4(ρ)

c(ρ) (3.4.7)

Cmc(r, ρ) = Fcy(ρ)−[xsc(r)− xcg(ρ)]Mt0(ρ) (3.4.8)

and the subscript sc denotes the shear center location. The shear center is assumedto be collocated with the center of gravity at the geometric centroid. The locations ofthese forces and moments are shown in figure 3.10.

The elastic twist is then

βe(r) =r∫

r0

Mt(ρ)K(ρ)Gdρ (3.4.9)


Figure 3.10: Force and moment locations for a blade element

To be rigorous, this elastic twist value should be added to the intrinsic blade twistβ and the loading then recomputed; then a new value of βe can be calculated and thisprocess iterated until convergence. However, because the purpose of this calculation isto verify that the elastic twist is small (|βe| < 1◦), an initial twist calculation meetingthis criteria will be assumed to result in a converged twist also of adequately lowmagnitude.

The maximum shear stress in the section is approximated by

τxy = MtC/K


where [28]

C = D

1 + π2D4

16A2

1 + 0.15(π2D4

16A2 −D

2rc

) (3.4.10)

D is the diameter of the largest inscribed circle in the section and can be approximatedby D = max(t). rc is the maximum radius of curvature of the section surface at thelocation of the largest inscribed circle.

The principal stresses σ1, σ2, and σ3 are τxy, −τxy, and σz in order of decreasingvalue. Then

τmax = |σ1 − σ3|/2

and σmax takes the value of the principal stress with the largest magnitude.The design stresses are taken to be [18]

σd = sy/2

τd = sy/4

where sy is the tensile yield strength. The design is considered acceptable when|τmax| < τd and |σmax| < σd for all blade stations.

By setting equation (3.4.2b) to zero, the rake angle that minimizes bending stressesvia centrifugal stiffening may be calculated by

θ = arcsin ρ

ρm

V ∗b2c∗Cl

2A∗r∗

(3.4.11)

Because, as implemented, blade-element momentum theory does not account forrake, this rake angle may be adopted with no impact on the performance calculations.However, because these structural analysis equations assume a small rake angle — notto mention that larger rake angles will affect the real-world performance — this angleis artificially limited to less than 10 degrees. For the propeller designs considered here,the rake angle prescribed by equation (3.4.11) will be significantly less than 10 degreesfor all of the blade span except the outer few percent.



Geometric Properties

To determine the geometric properties (cross-sectional areas, second moments of area,etc.), first the perimeter of the blade at each section is formulated as a series of points,taking into account chord, rake, twist, and sweep. Then A, Ix, Iy, and Ixy, and thecentroid location are calculated as described previously. The value of xcg is found byprojecting the centroid onto the chord line.

To calculate F from equation (3.4.1), each blade section is divided into an upperand lower surface, and the points comprising theses surfaces are interpolated such thatthe y coordinates for the points are given at the same x coordinates for both surfaces.The thickness t for each point can then be calculated as the distance between theupper and lower points. Using these new interpolated points, the camber line canbe calculated as the line through the averages of the upper and lower y coordinatesat each x location. The length of each camber line segment ∆Ui can then be easilycalculated, and the total arc length of the camber line U is used to calculate K.

To find the value of rc in equation (3.4.10), first the location of the maximumthickness is found (the thickness vector t having been previously calculated). Thenrc is found by using MATLAB’s FMINSEARCH optimizer to locate the two circles(one for the upper surface and one for the lower surface) that contain the point onthe surface at the location of the maximum thickness and the two adjacent perimeterpoints; rc is the radius of the larger of these two circles. Once rc is found, calculationof C is straightforward.

Forces and Moments

First, the x and y coordinates required for equations (3.4.6) are determined fromthe blade geometry. The rake angle is determined by the tangent of the slope of they-coordinates of the centers of pressure of the blade segments.

Next, the local torsionMt is calculated via numerical integration of equation (3.4.5),and the elastic twist is calculated via numerical integration of Mt using equation(3.4.9).


The distributed loads are calculated and the bending moments are calculated bynumerically integrating equations (3.4.3) and (3.4.4). Finally, the maximum shearand bending stresses per element are determined as outlined in the previous section.


Number of Chordwise Panels

Increasing the number of chordwise panels has a trivial effect on runtime givenreasonable values. Therefore, based on figure 3.11, Np will be set to 512 for thesecalculations.

Figure 3.11: Error in aeroelastic constraints relative to Np = 1024 (Ne = 512)


Figure 3.12 shows that increasing the number of elements is advantageous up untilNe = 1024, although with diminishing returns. Based on these results, Ne is chosento be 512 for these calculations.


Figure 3.12: Error in aeroelastic constraints relative to Ne = 4096 (Np = 151)

Chapter 4

Computational VerificationMethods

4.1 Analysis Using ANOPP PAS

The acoustic analysis is verified by comparison with a NASA production noise code,the Aircraft Noise Prediction Program Propeller Analysis System (ANOPP PAS).ANOPP PAS also uses BEM theory to compute propeller performance and Farassat’sFormulation 1A to compute acoustics. It can be configured to employ either thecompact chord formulation or the full blade formulation of the acoustics equations,facilitating the evaluation of the applicability of the compact chord approximationat this design point. ANOPP parameters used in this study are provided in ap-pendix A, and the blades are discretized into 21 stations using the spacing defined byequation (3.2.18).

4.1.1 Performance

Figure 4.1 compares the loading distributions of the 3- and 9-blade propellers describedin section 6.2.3 calculated by ANOPP PAS with the loading distributions calculatedusing the design method of section 3.2. The integrated loadings are given in table 4.1;ANOPP consistently predicts slightly higher thrust and power. Because ANOPP PAS

50

CHAPTER 4. COMPUTATIONAL VERIFICATION METHODS 51

cannot make calculations at J = 0, it was run at J = 0.001.

Method B CT CPANOPP PAS 3 0.2409 0.1147Section 3.2 3 0.2175 0.09879ANOPP PAS 9 0.2359 0.1131Section 3.2 9 0.2175 0.09821

Table 4.1: Comparison with ANOPP PAS integrated loading calculations

4.1.2 Acoustics

Figure 4.2 shows a comparison of the first two harmonics at 4R and θo = 90◦ withANOPP PAS compact chord and full blade formulation calculations for the 3-bladeand 9-blade geometries described in section 6.2.3, produced at J = 0.05 (V ∗∞ = 0.0159).First, ANOPP PAS was executed to analyze the performance and calculate theacoustics with the full blade formulation. Next, the loading distribution calculatedby this ANOPP PAS run was used to obtain the results from both the currentmethodology and ANOPP PAS using the compact chord approximation. For allconditions in these plots, the three analysis methods vary only trivially, with thecurrent methodology producing results between the two ANOPP methods in mostcases. Directivity patterns for MT = 0.36 are shown in figure 4.3.

4.2 Structural Analysis using XROTOR

XROTOR is a program for design and analysis of propellers and windmills whichemploys a three-axis nonlinear slender beam model to compute structural loads anddeflections. [7] To verify the structural analysis described in section 3.4, the XROTORexample case, an aquatic propeller described in table 4.2, was analyzed using bothXROTOR and the current analysis methods. Because XROTOR relies on specifiedmaterial properties and does not calculate stresses, only the calculation of Mt, βe,Sx, Sy, Mx, and My can be verified. The results are shown in figure 4.4; althoughXROTOR uses a substantially different method to compute these results, they are


(a) Thrust, 3-blade propeller (b) Power, 3-blade propeller

(c) Thrust, 9-blade propeller (d) Power, 9-blade propeller

Figure 4.1: Comparison with ANOPP PAS loading calculations at MT = 0.36

still quite similar to the calculations using the methodology of section 3.4. Thestair-stepping in figure 4.4b is a result of the limited precision given in XROTORoutput.


B 2σ 0.110CT 0.0749CP 0.190J 2.25MT 0.00583R 0.11 mRPM 759

Table 4.2: XROTOR example case

(a) Thickness noise, 3-blade propeller (b) Loading noise, 3-blade propeller

(c) Thickness noise, 9-blade propeller (d) Loading noise, 9-blade propeller

Figure 4.2: Comparison with ANOPP PAS acoustic calculations (first two harmonicsat 4R and θo = 90◦)


(a) Thickness noise, 3-blade propeller (b) Loading noise, 3-blade propeller

(c) Thickness noise, 9-blade propeller (d) Loading noise, 9-blade propeller

Figure 4.3: Directivity comparison at 4R with ANOPP PAS acoustic calculations(first harmonic, MT = 0.36)


(a) Local torsion (b) Aeroelastic twist

(c) Shear force (d) Moments

Figure 4.4: Comparison of structural calculations between the present methodologyand XROTOR

Chapter 5

Experimental Validation Methods

5.1 Propeller Manufacturing

To minimize the time and cost of producing small numbers of propellers, various rapidprototyping solutions — namely additive layer manufacturing technologies such asselective laser sintering (SLS), stereolithography (SLA), and fused deposition modeling(FDM) — were considered. The layer thickness for these processes typically variesbetween 0.003 and 0.005 inches (80-130 microns) for SLS and SLA but can be as lowas 16 microns (0.0006 inches) for FDM. This finite layer thickness can significantlydegrade the accuracy of the manufactured part if the part dimensions are close enoughto this layer size.

Each of these processes allows the use of a different set of manufacturing materials.The selection of a material (and, by extension, a process) depended greatly on thematerial strength and stiffness. Tensile modulus E and yield strength sy for variousavailable SLS, SLA, and FDM materials are shown in figure 5.1. Because Poisson’sratios and shear moduli for these materials were often not provided, the shear modulus,when unavailable, was estimated from the tensile modulus assuming a worst-casePoisson’s ratio of 0.5, which is not absurdly conservative since Poisson’s ratio for thesematerials is typically in the range of 0.42-0.46.

Although the Accura Bluestone material is exceptionally stiff, it is also unacceptablybrittle. Windform XT, although also quite strong and stiff, is about 60% more

56

CHAPTER 5. EXPERIMENTAL VALIDATION METHODS 57

Figure 5.1: Tensile yield strength and stiffness of select rapid prototyping materials

expensive than the next stiffest material, PA 3200 GF. Therefore, because PA 3200GF still provides adequate stillness, it was selected as the manufacturing material.

PA 3200 GF, produced by EOS, is a glass-filled polyamide powder. Relevantmaterial properties are given in Table 5.1 [8]. The layer thickness using this materialis 0.004 inches (100 microns).

Tensile modulus 3.3 GPaTensile strength 45 MPaDensity 1.25 g/cm2

Shear modulus 1.1 GPa (estimated)

Table 5.1: PA 3200 GF material properties

The manufactured geometry was measured using a coordinate measuring machine;typical results are shown in figure 5.2. Only the Q3 propeller was measured, due toblade deflections induced on the Q9 propeller by the minimum required measurementforce of the measuring probe. The geometry is quite accurately produced, and therounded trailing edge is likely a result of manual blade surface sanding as described in


section 6.3.

Figure 5.2: Comparison of intended section geometry with measured manufacturedgeometry at r∗ = 0.8 for a Q3 blade

5.2 Experimental Methods

Testing methods and apparatus were developed for the experimental evaluation ofpropeller designs and validation of the design and analysis methods. These apparatuscomprise a means to control the throttle and determine thrust, power, RPM, andnoise. A 14-pole brushless DC electric motor is mounted on linear bearings whichslide against a load cell to measure thrust, and a microphone with cardioid directivityrecords the noise. The microphone was calibrated using a sound level meter. Toreduce noise sources, the propeller is attached directly to the motor, eliminating theneed for a gearbox. At the relevant power range, the motor has an efficiency of 88%;this value can be used to calculate the power absorbed by the propeller by measuringthe power into the motor. All data from the test apparatus, including motor RPM,power, thrust, and microphone output, are simultaneously recorded digitally by asingle computer. The motor is powered by lithium-polymer batteries, and desiredRPMs are obtained by manually adjusting the throttle.

A commercial propeller, the four-bladed APC LP415512 of 15.5-inch diameter and


12-inch pitch, was employed to validate the test apparatus. The blade geometry wasprovided courtesy of APC, allowing the analysis method described in section 3.2.2 tobe used to predict the loading distributions. These loading predictions are then usedto predict the acoustics.

Test Stand 1, shown in figure 5.3, oriented the propeller axis horizontally onemeter from the ground such that the downwash impinged on the test stand. Acousticresults from this test stand, operating in a large indoor room, differed significantlyfrom expectations. An example spectral plot of the sideline (θo = 90◦) noise at 0.88meters (4.45R) recorded using Test Stand 1 is shown in figure 5.5a. The dominantharmonic is f1, not the expected blade passage frequency f4.

Figure 5.3: Test Stand 1 with the APC LP415512 propeller

Building on the experience operating this test stand, Test Stand 2 (shown infigure 5.4) was created. Test Stand 2 orients the axis of the propeller vertically suchthat the propeller thrusts downwards and the downwash is directed upwards, away


Figure 5.4: Operation of Test Stand 2

from the test stand. The propeller plane is located 2.7 meters above the ground toavoid recirculation and echoes. As shown in figure 5.5, the acoustic results from thisapparatus, operating outdoors in still air, are much closer to expectations, with theharmonic at the blade passage frequency within 4 dB of analytical predictions andthe harmonic noise lower relative to the tonal noise compared to the results from TestStand 1.

Performance results from experiment (using Test Stand 2) and analysis for thispropeller are shown in figure 5.6. Because this propeller is designed for forward flight,it is operating at high angles of attack at static conditions, making accurate analyticalperformance predictions difficult. However, this method predicts the thrust and powerwithin 9-20% and significantly outperforms ANOPP PAS.


(a) Test Stand 1, 4.45R

(b) Test Stand 2, 7.74R

Figure 5.5: Sideline noise results from Test Stands 1 and 2 with the LP415512 propeller

5.2.1 Discussion of Experimental Error

Determination of Thrust

Thrust is measured with a MSI Sensors FC22-3 compression load cell, which has a rangeof 25 lbf and a manufacturer-stated non-linearity, hysteresis, and nonrepeatability ofwithin ±1% of the output voltage. The analog output of this sensor is passed through aPhidgets PhidgetInterfaceKit 8/8/8 analog-to-digital converter, which reports the valueas an integer between 0 and 1000. At a specified throttle setting, minor fluctuations inrealized RPM add instability to the thrust reading. Experimental results have showna standard deviation in measured thrust at a fixed throttle setting of less than 2%(e.g. 1.1 N at 62 N thrust).


(a) Thrust (b) Power

Figure 5.6: Comparison of experimental performance results for the LP415512 propeller

Determination of Required Power

Power delivered to the motor is measured with an Eagle Tree Systems eLogger V4,which directly measures the voltage and current passing from the batteries to the motor.These power readings are fairly constant at constant throttle, with a standard deviationin reported power of less than 0.2% (e.g. 1.1W at a mean of 870W) demonstrated inexperiments. The efficiency is only known to two significant figures (88%± 0.5%), sothe estimate of power absorbed by the propeller inherits this level of accuracy.

Determination of Propeller RPM

RPM is determined with an Eagle Tree Systems Brushless Motor RPM Sensor V2,which measures the pulses on wires connecting the motor to the speed controller. Dueto the operation of brushless DC electric motors, this RPM reading is quite precise.At a fixed throttle setting, the standard deviation of the RPM at a mean RPM of 5500is 22, or 0.40%, allowing a desired RPM to be reached with this amount of precision.

Sound Measurement

The sound is measured with a Blue Microphones Yeti microphone that was calibratedwith a Tenma 72-940 sound level meter. The microphone has a frequency responseof 20 Hz – 20 kHz and a sensitivity of 4.5 mV/Pa (at 1 kHz) and is operated in its


cardioid directivity setting. The meter has a manufacturer-stated accuracy of ±1.5dB, a frequency range of 31.5 Hz – 8 kHz, and omnidirectional directivity. Duringcalibration, the microphone was operated in its omnidirectional directivity setting tobetter correlate with the meter readings, and the difference in manufacturer-statedfrequency response between the cardioid and omnidirectional directivity settings of themicrophone was applied to the calibration results. The manufacturer-stated frequencyresponse data is only available in digital graphical form with a minimum resolution of0.09 dB. Therefore, the calibrated accuracy of the microphone is about ±1.6 dB.

Chapter 6

Design Examples and Comparisonswith Experiment

6.1 Introduction

Two different propellers have been designed using the methodology developed inChapter 3. The results of experimental testing provide significant insight into theeffectiveness of this method at this design point and validate this design process asa means to develop low-noise electric UAV propellers. The two propeller designsfundamentally differ only in the blade count; all other relevant design variables (tipspeed, radius, section geometry, etc.) are held constant, and the same design methodis employed. Producing these two propellers allows the experimental verification ofthe effect of blade count on harmonic noise predicted in section 2.3.3.

6.2 Propeller Design Specifications

6.2.1 Design Parameters

As discussed in section 2.2, the flight speed was chosen to be V∞ = 0 and the radiuswas fixed at 9 inches (0.229 m). From the analysis in section 2.3.2 and 2.3.3, the tipspeed was chosen to be Mach 0.36 and the blade number was chosen to be 9. To

64

CHAPTER 6. DESIGN EXAMPLES AND COMPARISONS WITH EXPERIMENT65

verify the calculated harmonic noise decrease resulting from increased blade number,a three-bladed propeller was also chosen to be built and tested. The three- andnine-bladed propellers will be respectively referred to as Q3 and Q9.

6.2.2 Airfoil Selection

Because of the finite (0.004 inch) layer thickness of the chosen manufacturing methodand material, thicker sections are desired to minimize the deviation between theintended and realized geometry. Thicker sections are also beneficial for structuralconsiderations, and as discussed in section 2.3.6, the performance and acoustic penaltiesfor thicker sections are minor at the design point. Therefore, the section thicknesswas fixed at a relatively thick 15%, increasing to 20% from r∗ = 0.40 to r∗ = 0.25to reduce bending moments where local aerodynamic and aeroacoustic performanceis relatively unimportant. Figure 6.1 compares the current thickness specificationwith various other propellers: the Hartzell F8475 D-4 [19] used on the Piper CherokeeLance, the experimental 5868-9 propeller from the 1930s [13], the Branum-Tungmodel representing a typical tilt-rotor blade [4], the Dowty Rotol R212 [27] used onthe HS.748 airliner, and three APC model airplane propellers: 13× 6.5E, 15.5× 12(LP415512), and 27× 13E (where the “E” suffix indicates thinner blades intended forelectric motors).

Several common propeller airfoil geometries, thickened to 15% when necessary, wereanalyzed. These geometries are shown in figure 6.2 and the computed performance ofoptimal propeller designs using these different airfoils is shown in figure 6.3. At fixedthickness and for a point design, it is apparent that, at least for these conditions, theairfoil choice (given reasonable airfoils) is almost irrelevant, with the range of valuesbetween these seven airfoils only 1.3 dB and 0.009 FM .

Given these results, the ARA-D 10 airfoil was chosen, with the thickness scaled tomatch the prescribed thickness distribution.


Figure 6.1: Comparison of the current thickness specification to other propeller designs

6.2.3 Resultant Designs

Table 6.1 and figure 6.5 summarize the resulting 3- and 9-blade propeller designs. Thethree-blade propeller was designed for maximum thrust given a power of 1500 W; thenine-blade propeller was designed to minimize power while producing the same thrustas the three-blade design. The nine-blade propeller is slightly more efficient, due tolower tip losses, but it experiences more aeroelastic twist and higher shear and normalstresses.

From figure 6.6, it can be seen that thickness noise dominates in the forwardhemisphere and loading dominates in the aft hemisphere. If thinner sections couldbe used, lower thickness noise could reduce the total acoustic energy by up to a fewdecibels.


ARA-D 10

ARA-D 13

Clark Y

NACA 0015

NACA 2415

NACA 4415

RAF 6

Figure 6.2: Airfoil section geometries, scaled to 15% thickness

B 3 9T 84.68 N 84.68 NP 1500 W 1491 WCT 0.2175 0.2175CP 0.09879 0.09821FM 0.8192 0.8420σ 0.2877 0.2684Thickness noise 69.9 dB (61.5 dBA) 19.7 dB (18.7 dBA)Loading noise 68.2 dB (59.8 dBA) 21.7 dB (20.7 dBA)Total noise 72.7 dB (64.3 dBA) 24.1 dB (23.2 dBA)

Table 6.1: Propeller design specifications

6.3 Comparison of Acoustic Energy and Total Thrustand Power

To evaluate the relative performance of these designs, the noise and efficiency of thetwo propellers are compared to a similar commercial propeller, the four-blade APC15.5× 12 (LP415512). Figure 6.7 compares the analytical and experimental sidelinenoise results for the Q3, Q9, and APC 15.5× 12 propellers. As built, the Q3 and Q9propellers possessed a fair amount of surface roughness; the Q3 propeller was analyzedboth before and after sanding the blades smooth with 400 grit sandpaper.


Figure 6.3: Predicted sensitivity of efficiency and noise of an optimum propeller toairfoil choice (T = 84 N, MT = 0.36, R = 0.289m, V∞ = 0)

The 90 Hz tone in these plots does not originate from the propellers and can beignored. The significant tones around 4000 Hz are assumed to originate from themotor and can be ignored as well. The tone at f1 may also be at least partiallyproduced by the motor.

Notably, the blade passage frequency tone diminishes from 87 dB for the Q3propeller to 58 dB for the Q9 propeller. Although the predicted Q9 blade passagefrequency SPL of 30 dB is not realized — likely due to imperfect phase cancellationresulting from geometric differences between the different blades caused by finitemanufacturing tolerances – this decrease is still quite significant. The tone at f3, theblade passage frequency of Q3, is diminished from 87 dB for the Q3 propeller to 55dB for the Q9 propeller. Also notable is the increase in high-frequency broadband


(a) 3-blade design (b) 9-blade design

Figure 6.4: Propeller geometry (planform view)

noise in Q9 relative to Q3. These results validate the assumption that broadbandnoise is relatively unimportant compared to harmonic noise at these conditions.

Comparison of the rough and smooth results for Q3 show that the surface roughnesshas a trivial impact on harmonic noise but increases broadband noise by a smallamount (on the order of 5 dB).

Analysis of the APC 15.5×12 at the same tip Mach number confirms the expectationthat the blade passage frequency tone would be quieter than Q3 but louder than Q9:the measured value is 71 dB, compared to 87 dB for Q3 and 58 dB for Q9.

Figure 6.8 shows that the experimental thrust and figures of merit are somewhatless than the predicted values for both the Q3 and Q9 propellers, although the poweris as predicted for the higher tip Mach numbers. This discrepancy is partly expected,due to the choice of performance model; further potential explanations are discussedin section 7.1.


(a) Chord (b) Twist (c) Lift coefficient

(d) Thrust (e) Power (f) Reynolds number

(g) Rake (h) Aeroelastic twist (i) Stress

Figure 6.5: Specifications and predicted performance of the Q3 (blue) and Q9 (green)propellers


(a) Q3 (b) Q9

Figure 6.6: Predicted OASPL directivity patterns


(a) Q3 (rough)

(b) Q3 (smooth)

(c) Q9 (rough)

(d) APC 15.5× 12

Figure 6.7: Sideline noise at 6R and MT = 0.32


(a) Thrust

(b) Power

(c) Figure of Merit

Figure 6.8: Experimental (solid lines) and analytical (dashed lines) performancecomparison

Chapter 7

Conclusions

Developing a methodology for the design of a quiet propeller for an electric UAV atstatic conditions presents challenges in striking a balance between fidelity and projectscale. The goals of this research have been to develop a practical design methodologygiven these challenges and to evaluate the effectiveness of this methodology throughthe design, manufacture, and testing of new quiet propeller models.

7.1 Summary of Results and Contributions

The experimental performance results of the Q3 and Q9 propellers, as presentedin figure 6.8, illustrate a significant difference between the predicted and measuredperformance. Although a performance discrepancy is expected at static conditions dueto the wake geometry assumptions of blade-element theory, the observed discrepancycould also be a result of:

• Three-dimensional blade aerodynamic and rotational effects not predicted byblade-element momentum theory that effectively modify airfoil section perfor-mance

• Aeroelastic effects not predicted by the model described in section 3.4

• Experimental error, as explained in section 5.2.1; however, certain sources of

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CHAPTER 7. CONCLUSIONS 75

experimental error can be ruled out, because multiple experimental runs atdifferent instances produced consistent results

Because of this discrepancy, the analytical performance results must be interpretedwith some skepticism. This discrepancy also reinforces the difference in results betweenblade-element momentum theory and higher-fidelity methods such as Euler or Navier-Stokes simulations. While employing such high-fidelity methods would have, in alllikelihood, resulted in much better agreement between analysis and experiment, thetime required to achieve a usable design with these methods (including the process ofproducing computational meshes) is extremely high, rendering this option impractical.

However, the experimental acoustic results show that the chosen performanceanalysis method did result in a design methodology useful in evaluating noise-reductionmethods. The analytical noise analysis performed investigated a broad spectrum ofpotential noise reduction methods: lowering the tip speed and increasing the bladecount were chosen as the most promising methods, while sweep, unequal blade spacing,thickness reduction, and split tips were shown to be ineffective at these conditions.

The noise reduction effect of increasing blade number was quite successfullydemonstrated experimentally. The noise reduction did not reach the extreme levelspredicted by the analysis method, but, as stated in section 6.3, this is likely a resultof manufacturing error precluding perfect phase cancellation between the differentblades, and the noise reduction achieved is certainly still significant. Additionally, thepredicted amplitude of the first harmonic of the propellers with fewer blades, and,therefore, less cancellation – i.e., the Q3 propeller and the APC 15.5× 12 – matchedthe experimentally observed values well.

The experimental acoustic analysis also reveals another important result: thatbroadband noise is noticeable but largely insignificant for these propellers at theseconditions. This result is difficult to predict analytically due to the lack of relevantempirical data.

CHAPTER 7. CONCLUSIONS 76

7.2 Consideration of Future Work

There are many areas in which continued effort could improve and expand upon themethods and analyses developed here. Foremost is the employment of a performanceanalysis method more accurate at static conditions. Higher-fidelity methods bettersuited to these conditions, such as Navier-Stokes analysis, were beyond the scope ofthis project, but could be employed in the future along with the aeroacoustic andstructural analysis methods described above.

A higher-fidelity structural model, such as a finite-element method, would increasethe complexity of the analysis but would allow smaller factors of safety and couldmodel a greater variety of blade construction configurations, such as those morerepresentative of a production blade.

Along this vein, the propellers could be constructed in a method representative ofproduction propellers. This could necessitate a different structural analysis method,since the current method assumes a homogeneous construction. The design method-ology could also be applied to propellers of different sizes and thrust ratings toexperimentally verify the applicability of these results to a variety of configurations.

Further increasing the complexity and fidelity of the methodology, aerodynamicand aeroacoustic interactions between the propeller and the aircraft structures couldbe studied, allowing the development of a quiet efficient propeller optimized for aspecific aircraft installation. Wind tunnel and/or flight tests could provide experimentalanalysis capabilities at nonzero advance ratios, allowing the development and validationof a methodology that applies to a greater range of flight conditions.

Appendix A

ANOPP Parameters

Parameter Value DescriptionEPSILON 10−5 Error criterion for stopping the iteration on the induced veloc-

itiesEPSLON 10−6 Blade aerodynamics convergence limit: the conformal mapping

iterative solution is completed when the residual is less thanor equal to EPSLON

FLAT false Flag to select zero pressure gradient flat-plate modelHSEP 0.737 Value of turbulent boundary-layer modified shaped factor at

the location of separationIATM 0 Atmospheric conditions selector (0 = a0 and ρ are input vari-

ables; 1 = conditions are read from a provided data table)ICL 1 Compressibility correction of lift coefficients selector flag (0 =

no correction; 1 = Glauert’s correction)ICP 1 Compressibility correction of pressure coefficients selector flag

(0 = no correction; 1 = Glauert’s correction; 2 = KarmanTsien’s correction)

IDPDT 0 Blade loading selector (0 = steady; 1 = time dependent)INDUCED false Induced drag option (false: total drag = profile drag + induced

drag; true: total drag = profile drag)

77

APPENDIX A. ANOPP PARAMETERS 78

NTIME 768 Number of time points used for time signature in acousticscalculations

NWEIGHT 0.1 Weighting factor for Newton iteration in the propeller perfor-mance module

OPTION 2 Performance methodology option (0 = blade-element momen-tum theory; 1 = Prandtl tip relief correction, chord goes tozero at blade tip; 2 = Prandtl tip relief correction, chord doesnot go to zero at blade tip; 3 = modified Prandtl tip lossfunction based upon empirical studies on prestall conditions)

THETAR 0 Root blade pitch setting (radians)

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