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Aerodynamic Propeller Model

for Load Analysis

M A R I O H E E N E

Master of Science Thesis Stockholm, Sweden 2012

Aerodynamic Propeller Model for Load Analysis

M A R I O H E E N E

Master’s Thesis in Scientific Computing (30 ECTS credits) Master Programme in Scientific Computing 120 credits

Royal Institute of Technology year 2012 Supervisor at KTH was David Eller

Examiner was Michael Hanke TRITA-MAT-E 2012:02 ISRN-KTH/MAT/E--12/02--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

AbstractAn aerodynamic propeller model, which can contribute tothe prediction of structural loads experienced by aircraft indifferent flight maneuvers is presented.

The model is based on Blade Element Momentum theoryand is able to predict the unsymmetrical and frequency-dependent forces and moments induced by the propelleron the airplane structure at steady and unsteady inflow-conditions.

In order to validate the model, a comparison with exper-imental results was performed and it can be seen that themodel is in agreement with the experimental data providingthat the aerodynamic data used for the calculations hasgood accuracy.

ReferatAerodynamisk propellermodell för

simuleringsbaserade lastberäkningar

En modell har utvecklats för att beräkna aerodynamiskakrafter som orsakas av propellern vid manöverflygning. Mod-ellen använder sig av klassiska bladelementteorin för predik-ering av osymmetriska stationära krafter som uppstår vidsnedanblåsning av propellerskivan. Modellen kommer attanvändas inom ett forskningsprojekt om effektiv beräkningav aerodynamiska laster vid flygmanövrar och i vindbyar.En vidareutveckling av den klassiska metoden används föratt ta fram instationära kraftbidrag i frekvensplanet i enform som är lämpligt för aeroelastiska stabilitetsanalys ochberäkning av vindbylasterna.

Jämförelser med omfattande experimentella resultat hargenomförts för att validera modellen. Inom ramen för anta-ganden och noggrannheten i modellens indata kan modellenstillförlitighet bedömas som tillräckligt för ändamålet. Däre-mot visar sig att propellermodellen är – som förväntat –mindre lämpligt för att bedöma propellerlasterna utanförpropellerns reguljära driftområdet.

AcknowledgementsFirst and foremost, I would like to express my gratitude tomy supervisor, David Eller, from the Division of Flight Dy-namics at KTH Stockholm, for his valuable advice through-out all stages of my thesis.

I also thank Michael Hanke, from KTH Stockholm, andUlrich Rüde and Harald Köstler, from the University ofErlangen-Nuremberg, for their assistance concerning thedouble-degree Master’s program.

I wish to thank Jesper Oppelstrup, from KTH Stock-holm, for his hospitality and all the interesting discussionsat our lunch meetings.

Furthermore, I am very grateful to my parents for theirsupport throughout all steps of my education.

Special thank goes to the Erich-Becker-Stiftung, whosefinancial support for my thesis has been a great help for me.

Contents

1 Introduction 1

2 Theory of propeller modelling 32.1 Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Combined Blade Element Momentum Theory . . . . . . . . . . . . . 92.4 Non-dimensional formulation . . . . . . . . . . . . . . . . . . . . . . 112.5 Unsteady propeller model . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Steady propeller model with oscillatory forces . . . . . . . . . . . . . 15

3 Implementation 173.1 Solution of the governing BEM equations and evaluation of section

force and moment coefficients . . . . . . . . . . . . . . . . . . . . . . 173.2 Adaptive integration of force and moment coefficients . . . . . . . . 193.3 Representation of geometry and aerodynamic data . . . . . . . . . . 21

4 Validation 254.1 Parallel inflow-conditions . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Non-parallel inflow-conditions . . . . . . . . . . . . . . . . . . . . . . 30

5 Conclusion 35

Bibliography 37

A Jacobian matrix for the Newton solver 39

B Linearization of the BEM equations with respect to α 41

C Linearization of the BEM equations with respect to β 43

D Code documentation 45

Chapter 1

Introduction

Prediction of flight loads plays an important role in the design process of aircraft.This requires models to describe the structural loads due to aerodynamic forces andaeroelastic effects with sufficient accuracy, but low enough computational effort inorder to be feasible for practical application.

This work describes the theory, implementation and validation of an aerodynamicpropeller model based on the Blade Element Momentum Theory. Applying themodel in flight simulation, the forces and moments induced by the propeller onthe airplane structure in different flight maneuvers can be analyzed and serve asbasis for structural load analysis. Furthermore, the model can be applied to predictthe frequency-dependent forces for unsteady inflow-conditions, as they occur dueto structural vibrations of the wing on which the propeller is mounted. This is ofinterest for gust load analysis and fatigue failure predictions.

The Blade Element Momentum theory combines Blade Element Theory withMomentum Theory. Blade Element Theory dates back to the work of Drzewiecki [6]and is based on the idea of predicting the performance of a given propeller geometry byconsidering the aerodynamic forces produced by the two-dimensional blade sections.Momentum Theory was initially developed in the 19th century by Rankine [14] andFroude [7] for marine propellers. They described the important relation betweenthrust and acceleration of the airflow by the propeller. The Wright brothers were thefirst to combine Blade Element Theory and Momentum Theory in order to designairplane propellers [17]. The corrections to the theory presented by Goldstein [9]and Prandtl [13] were an important contribution in order to improve the accuracyof the theory.

Aerodynamic forces

This section will provide a short introduction to the basic principles and definitionsin aerodynamics. It is based on the textbook by Anderson [1].

The geometry of every body being exposed to a fluid flow causes a certainpressure distribution along its surface. The shape of a wing, usually, causes a lower

1

CHAPTER 1. INTRODUCTION

pressure along its upper surface than on its lower surface, resulting in a force thatpushes the wing upwards.

L

R

D

N

A

V ∞

α

c

Figure 1.1. Aerodynamic forces acting on an airfoil.

The forces acting on an airfoil are sketched in Figure 1.1. V∞ denotes the velocityand direction of the freestream, e.g. the flow far away from the body. The force Racting on the airfoil is composed by the lift force L, acting perpendicular to V∞,and the drag force D, acting parallel to V∞. The chord line c is defined as the lineconnecting the leading edge and the trailing edge of the airfoil. The normal forceN , perpendicular to c, and the axial force A, parallel to c, can be expressed by thefollowing geometrical relation:

N = L cosα+D sinα (1.1)A = −L sinα+D cosα (1.2)

It is common to express lift and drag by the non-dimensional coefficients

cL = L12ρ∞V

2∞c

(1.3)

cD = D12ρ∞V

2∞c

, (1.4)

where ρ∞ denotes the density of the freestream. The variation of cL and cD withrespect to α is a very important characteristic of every airfoil.

2

Chapter 2

Theory of propeller modelling

2.1 Blade Element TheoryThe Blade Element Theory (BET) is a relatively simple model to predict theperformance of a certain propeller geometry. It was initially presented by W.Froude [7], in 1878, and later refined by Drzewiecki [6]. In BET the propellerblade is subdivided in small elements and the two-dimensional flow around eachelement is analyzed individually. The theory is based on the assumption that nointerference between adjacent blade elements occurs, which is, however, withoutphysical justification [17]. The representation of Blade Element Theory used in thiswork was adapted from the textbooks of Glauert [8] and Weick [18].

Figure 2.1. Propeller blade with single blade element (adapted from [18]).

As shown in Figure 2.1, the distance from the propeller axis to the centerline ofeach element is denoted by r and the width of the element by dr. R is the distancefrom the propeller axis to the tip of the blade, also called tip radius. Every bladesection is further specified by its airfoil geometry, the chord length c, and the twistangle τ (see also Section 3.3).

The forces and velocities acting at the blade element are visualized in Figure 2.2.The angle θ between chord line and propeller plane is composed of the twist angleof the blade section and the pitch angle of the propeller blade. Here, the pitch angleof the blade is defined as the angle between the chord line of the tip section and

3

CHAPTER 2. THEORY OF PROPELLER MODELLING

Figure 2.2. Definition of forces and velocities acting at the blade element.

the propeller plane. The twist angle of the blade section is defined as the anglebetween the chord line of the blade section and the propeller plane, when the bladepitch angle is zero. The flow to which the airfoil is exposed is denoted by Vp andis composed of the axial velocity Vx and the tangential velocity Vt. The velocitycomponents will be examined in more detail later. The angle between Vp and thepropeller plane is denoted by

ϕ = tan−1 VxVt

(2.1)

and thus the local angle of attack of the airfoil is denoted by

αb = θ − ϕ. (2.2)

According to the definitions in Section 1, the elemental lift force dL and dragforce dD acting on the blade element can be formulated as

dL = 12ρV

2p cL(αb)cNbdr (2.3)

dD = 12ρV

2p cD(αb)cNbdr, (2.4)

where Nb is the number of propeller blades.The elemental thrust dT and torque dQ can be derived by the following geomet-

rical transformations:

dT = dL cosϕ− dD sinϕ (2.5)dQ

r= dL sinϕ+ dD cosϕ. (2.6)

As it will be discussed in more detail in Section 2.2, the propeller induces velocityin axial direction, which increases the flow speed at the propeller plane by va. The

4

2.1. BLADE ELEMENT THEORY

Figure 2.3. Components of the freestream velocity V∞.

propeller also causes a rotation in the flow field, in the same direction as the propellerrotates, and thus the tangential velocity seen by the blade element is reduced by vr.

The components of Vp can then be expressed as

Vx = V ′x + va (2.7)Vt = V ′t − vr. (2.8)

V ′x and V ′t depend on the angle of attack α and the sideslip angle β between thepropeller axis and the freestream velocity V∞. Additionally, V ′t depends on theangular position of the blade, the local radius r of the blade element, and the rotationspeed of the propeller ω.

From Figure 2.3 the following relations for the velocity components of V∞ canbe derived:

vx = V∞ cosβ cosα (2.9)vy = V∞ sin β (2.10)vz = V∞ cosβ sinα. (2.11)

The propeller axis is defined in the opposite direction than vx.The axial velocity V ′x is independent of the radius and the angular position of

the blade and thus for all blade elements holds

V ′x = vx = V∞ cosβ cosα. (2.12)

The tangential velocity V ′t depends on the angular position γ of the blade, asvisualized in Figure 2.4:

V ′t = ωr + vy sin γ − vz cos γ= ωr + V∞ sin β sin γ − V∞ cosβ sinα cos γ. (2.13)

5


Therefore, it is not only necessary to consider different radii along the blade, but alsoto perform the analysis at a number of different angular positions. Incorporatingthe dependency on γ in the above equations for thrust and torque

dT = dT (r, γ) = 12ρV

2p [cL(αb) cosϕ− cD(αb) sinϕ] cNbdr

dγ

2π (2.14)

dQ = dQ(r, γ) = 12ρV

2p [cL(αb) sinϕ+ cD(αb) cosϕ] cNbrdr

dγ

2π . (2.15)

Figure 2.4. Components of the tangential velocity.

Since the induced velocities, va and vr, are unknown, Blade Element Theoryon its own can only provide an upper limit for the performance of a propeller bysetting va and vr to zero. To predict the performance of a propeller more accuratelya relation for va and vr is necessary, as it is provided by Momentum Theory.

2.2 Momentum TheoryMomentum Theory was originally developed by Rankine [14] and Froude [7] and isbased on the idea that whenever a propeller produces thrust, it causes a motion ofthe air in opposite direction to the thrust [18]. This section is based on the textbooksby Glauert [8] and Weick [18].

In Momentum Theory the propeller is assumed to have an infinite number ofblades and thus can be understood as a circular disk, also called actuator disk inthis context. Further assumptions are that the flow is inviscid, incompressible anduniform over the whole disk. The Momentum Theory, as presented by Rankineand Froude, neglects any rotational motion in the slipstream and frictional dragon the propeller blades. The general concept of Momentum Theory is visualized in

6

2.2. MOMENTUM THEORY

Figure 2.5. Pressure and velocity in front of and behind the propeller disk (reprintedfrom [18]).

Figure 2.5. Far away in front of the propeller disk the flow has freestream velocityV∞ and pressure p∞. The airflow passes the propeller disk with velocity V . Thepressure is increased from p′ in front of the disk to p′ + ∆p behind the disk. Farbehind the disk, the flow has velocity V1 and the pressure decreased to the initialp∞.

Since the flow is assumed to be inviscid and incompressible, according toBernoulli’s equation it holds

p∞ + 12ρV

2∞ = p′ + 1

2ρV2 (2.16)

in front of the disk and

p′ + ∆p+ 12ρV

2 = p∞ + 12ρV

21 (2.17)

behind the disk. It follows that

∆p =[p∞ + 1

2ρV2

1

]−[p∞ + 1

2ρV2∞

]= 1

2ρ(V 21 − V 2

∞). (2.18)

The assumption of incompressibility simplifies the analysis, but other unique relationsbetween velocity and pressure, such as that of isentropic compressible flow, could beused as well. The trust T can be expressed as

T = A∆p = A12ρ(V 2

1 − V 2∞), (2.19)

where A denotes the area of the disk. The thrust also equals the increase inmomentum in the slipstream, which is the product of the mass flow through the diskAρV and the overall increase in velocity

T = AρV (V1 − V∞). (2.20)

7


Equating (2.19) and (2.20)

A12ρ(V 2

1 − V 2∞) = AρV (V1 − V∞), (2.21)

it followsV = V1 + V∞

2 . (2.22)

Defining the increase in velocity at the propeller disk as the axial induced velocityva, analogously to (2.22) it holds

V = V∞ + va (2.23)V1 = V∞ + 2va (2.24)

and thusT = 2Aρ(V∞ + va)va. (2.25)

In this simplified version of the Momentum Theory any rotational motion in theslipstream is neglected. However, in order to predict the performance of an aircraftpropeller more accurately, it is necessary to extend the actuator disk model by arotational component being imparted to the fluid flow as a reaction to the torque ofthe propeller.

Figure 2.6. Streamtube through an annular element of the propeller disk (adaptedfrom [12]).

Figure 2.6 shows the streamtube through an annular ring of the propeller diskwith radius r and width dr. The rotation speed of the flow inside the streamtube, atthe position of the propeller disk, is denoted by vr. The rotation speed far behindthe disk is denoted by vr,1. The elemental torque dQ equals the increase in angularmomentum in the streamtube

dQ = ρV dAvr,1r, (2.26)

where ρV dA is the mass flow through the annular ring.

8

2.3. COMBINED BLADE ELEMENT MOMENTUM THEORY

It can be shown (see, e.g., [8]) that for incompressible flow the rotational velocityat the disk

vr = 12vr,1 (2.27)

and thusdQ = ρV dA2vrr = 4πr2drρ(V∞ + va)vr. (2.28)

Analogously, the elemental thrust produced by the annular element of the propellerdisk

dT = 4πrdrρ(V∞ + va)va. (2.29)

The above equations, however, have been developed under the assumption oftreating the propeller as a circular disk, i.e., with an infinite number of blades. Thereexist solutions by Prandtl [13] and Goldstein [9] to take into account the effectsthat occur with a finite number of blades. Although Goldstein’s solution is moreaccurate, it is too complicated for general use. Therefore, Glauert [8] proposed thefollowing correction to Momentum Theory

dT = 4πrdrρ(V∞ + va)vaF (2.30)dQ = 4πr2drρ(V∞ + va)vrF, (2.31)

where

F = 2π

arccos e−f (2.32)

f = Nb

2R− rr sinϕ. (2.33)

As introduced in Section 2.1, Nb denotes the number of blades and ϕ is the inflowangle at the current blade element with respect to the propeller plane.

Incorporating the dependency on the angular position γ analogously to Section 2.1and considering only a segment of the annular ring

dT (r, γ) = 2rρ(V ′x + va)vaFdrdγ (2.34)dQ(r, γ) = 2r2ρ(V ′x + va)vrFdrdγ. (2.35)

2.3 Combined Blade Element Momentum TheoryIn the previous sections two different theories to formulate the elemental thrustdT and torque dQ have been presented. Combining Blade Element Theory andMomentum Theory allows to calculate the induced velocities, va and vr.

Equating the elemental thrust, (2.14) and (2.34),

12ρV

2p [cL(αb) cosϕ− cD(αb) sinϕ] cNbdr

dγ

2π = 2rρ(V ′x + va)vaFdrdγ (2.36)

9


Figure 2.7. Definition of forces and moments on the propeller disk.

and elemental torque, (2.15) and (2.35),

12ρV

2p [cL(αb) sinϕ+ cD(αb) cosϕ] cNbrdr

dγ

2π = 2r2ρ(V ′x + va)vrFdrdγ, (2.37)

results in a system of non-linear equations

f1(va, vr) = 12σV

2p [cL(αb) cosϕ− cD(αb) sinϕ]− 2(V ′x + va)vaF = 0 (2.38)

f2(va, vr) = 12σV

2p [cL(αb) sinϕ+ cD(αb) cosϕ]− 2(V ′x + va)vrF = 0, (2.39)

which can be solved for va and vr, e.g., with an iterative procedure like Newton’sMethod. The non-dimensional factor σ = cNb

2πr denotes the so-called blade solidity.After the induced velocities, va and vr, have been determined, the overall

performance of the propeller can be calculated by integration of the elemental thrustand torque over the propeller disk

T =∫ 2π

0

∫ R

RHub

dT (r, γ) (2.40)

Q =∫ 2π

0

∫ R

RHub

dQ(r, γ). (2.41)

RHub denotes the radius of the propeller hub or the root of the blade, respectively.The forces and moments acting on the propeller axis can be derived from the

10

2.4. NON-DIMENSIONAL FORMULATION

elemental thrust and torque, as shown in Figure 2.7:

dFx = dT dMx = −dQ

dFy = dQ

rsin γ dMy = dTr sin γ (2.42)

dFz = dQ

rcos γ dMz = dTr cos γ.

The fact that dMx is defined in the opposite direction than dQ, results from aright-handed coordinate system being used for the definition of forces and momentsacting on the propeller axis, where the x-axis points in the opposite direction thanthe vx-component of the freestream velocity (see Figure 2.2 and Figure 2.3). Theoverall forces Fx, Fy, Fz, and moments Mx, My, Mz can be determined from theelemental values by integration, as presented in Equation (2.40) and Equation (2.41).

2.4 Non-dimensional formulationUsually, it is not desirable to provide absolute values for the velocities when examiningthe performance of a propeller. Therefore, it is more common (see, e.g., [8, 18]) touse the non-dimensional rate of advance

λ = V∞ωR

(2.43)

orJ = V∞

nD= λπ, (2.44)

where n is the number of revolutions per second and D = 2R is the diameter of thepropeller. The other velocities can be replaced by non-dimensional expressions inthe same way:

λx = V ′xωR

= λ cosβ cosα (2.45)

λt = V ′tωR

= r

R+ λ(sin β sin γ − cosβ sinα cos γ) (2.46)

λa = vaωR

(2.47)

λr = vrωR

(2.48)

λ2p = (λx + λa)2 + (λt − λr)2. (2.49)

Inserting the above expressions in Equation (2.38) and Equation (2.39) resultsin the non-dimensional formulation of the Blade Element Momentum approach

f1(λa, λr) = 12σλ

2p [cL(αb) cosϕ− cD(αb) sinϕ]− 2(λx + λa)λaF = 0 (2.50)

f2(λa, λr) = 12σλ

2p [cL(αb) sinϕ+ cD(αb) cosϕ]− 2(λx + λa)λrF = 0, (2.51)

11


whereϕ = tan−1λx + λa

λt − λr. (2.52)

With non-dimensional coefficients for thrust

dCT = 2dTρ(ωR)2πR2 = 4r(λx + λa)λaF

R2πdrdγ (2.53)

and torque

dCQ = 2dQρ(ωR)2πR3 = 4r2(λx + λa)λrF

R3πdrdγ (2.54)

the following coefficients corresponding to the elemental forces and moments actingon the propeller axis can be formulated:

dCFx = dCT dCMx = −dCQ

dCFy = dCQR

rsin γ dCMy = dCT

r

Rsin γ (2.55)

dCFz = dCQR

rcos γ dCMz = dCT

r

Rcos γ.

Analogous to Equation (2.40) and (2.41), integration of the elemental coefficientsover the propeller disk yields the overall force CFx , CFy , CFz and moment coefficientsCMx , CMy , CMz .

2.5 Unsteady propeller model

The goal of the unsteady propeller model is to analyze variations in the force andmoment coefficients as a reaction to unsteady variations in the flow field, being,for example, caused by structural vibrations of the wing on which the propeller ismounted. The model can be used to analyze the frequency dependent loads that areinduced by the propeller disk on the airplane structure.

Since the variations in the α and β angles, caused by oscillations of the wingstructure, can be assumed to be small-amplitude deviations from the steady-stateoperation point, linearization can be applied to efficiently approximate the force andmoment coefficients. The derivation will be explained on the basis of variations in α,but is likewise valid for variations in β. Linearization of the governing equations ofthe blade element momentum theory, Equation (2.50) and (2.51), with respect to adeviation in α from the operation point yields

f1(α0 + ∆α) = f1(α0) + ∂f1∂α

∣∣∣∣α0

∆α = 0 (2.56)

f2(α0 + ∆α) = f2(α0) + ∂f2∂α

∣∣∣∣α0

∆α = 0, (2.57)

12

2.5. UNSTEADY PROPELLER MODEL

where α0 denotes the angle of attack in the operation point. As presented in theprevious sections, for the steady-state solution it holds

f1(α0) = 0 f2(α0) = 0

and thus it must hold

∂f1∂α

∣∣∣∣α0

= 0 ∂f2∂α

∣∣∣∣α0

= 0, (2.58)

which results in the linear system of equations

∂f1∂α

∣∣∣∣α0

=12σλ

2p

∂ct∂α

+ 12σct

∂λ2p

∂α− 2λaF

∂λx∂α− 2λaF

∂λa∂α

− 2(λx + λa)F∂λa∂α− 2(λx + λa)λa

∂F

∂α= 0 (2.59)

∂f2∂α

∣∣∣∣α0

=12σλ

2p

∂cq∂α

+ 12σcq

∂λ2p

∂α− 2λrF

∂λx∂α− 2λrF

∂λa∂α

− 2(λx + λa)F∂λr∂α− 2(λx + λa)λr

∂F

∂α= 0. (2.60)

Here,

ct = cL(αb) cosϕ− cD(αb) sinϕ (2.61)cq = cL(αb) sinϕ+ cD(αb) cosϕ, (2.62)

which can also be expressed as

ct = cZ(αb) cos θ − cX(αb) sin θ (2.63)cq = cZ(αb) sin θ + cX(αb) cos θ, (2.64)

by using the normal force and axial force coefficient

cZ = cL(αb) cosαb + cD(αb) sinαb (2.65)cX = −cL(αb) sinαb + cD(αb) cosαb. (2.66)

A detailed derivation of the BEM equations with respect to α and β and the resultinglinear system of equations can be found in Appendix B and C.

Solving Equation (2.59) and (2.60) for the unknowns ∂λa∂α and ∂λr

∂α , the derivativesof the elemental thrust and torque coefficient

∂dCT∂α

∣∣∣∣α0

= 4rR2π

[λaF

(∂λx∂α

+ ∂λa∂α

)+ (λx + λa)F

∂λa∂α

+ (λx + λa)λa∂F

∂α

]drdγ (2.67)

13


∂dCQ∂α

∣∣∣∣α0

= 4r2

R3π

[λrF

(∂λx∂α

+ ∂λa∂α

)+ (λx + λa)F

∂λr∂λa

+ (λx + λa)λr∂F

∂α

]drdγ (2.68)

can be determined. For the derivatives of the force and moment coefficients thesame relations as in Equation (2.56) hold

∂dCFx

∂α= ∂dCT

∂α

∂dCMx

∂α= −∂dCQ

∂α∂dCFy

∂α= ∂dCQ

∂α

R

rsin γ

∂dCMy

∂α= ∂dCT

∂α

r

Rsin γ (2.69)

∂dCFz

∂α= ∂dCQ

∂α

R

rcos γ ∂dCMz

∂α= ∂dCT

∂α

r

Rcos γ.

Integration over the propeller disk yields the derivatives of the overall force andmoment coefficients

∂CFj

∂α=∫ 2π

0

∫ R

RHub

∂dCFj (r, γ)∂α

(2.70)

∂CMj

∂α=∫ 2π

0

∫ R

RHub

∂dCMj (r, γ)∂α

, (2.71)

which can be used to approximate the force and moment coefficients for smalldeviations in α from the operation point

CFj (α0 + ∆α) = CFj (α0) +∂CFj

∂α

∣∣∣∣α0

∆α (2.72)

CMj (α0 + ∆α) = CMj (α0) +∂CMj

∂α

∣∣∣∣α0

∆α. (2.73)

Special treatment is necessary for the evaluation of ∂ct∂α and ∂cq

∂α . The normalforce coefficient cZ does not respond instantaneously to variations in the inflow angle,but experiences a certain phase shift, which is dependent on the frequency of theoscillation. Using the same assumptions as for the steady-state case, e.g. incom-pressible, inviscid and two-dimensional flow around the blade section, Theodorsen’ssolution for thin airfoils oscillating in incompressible flow (as presented in [2, 3]) canbe applied. Hence,

∂cZ(k)∂αb

= ∂

∂αb(cL cosα+ cD sinα)

(12 ik + C(k)

)=[(∂cL∂αb

+ cD

)cosα+

(∂cD∂αb− cL

)sinα

](12 ik + C(k)

). (2.74)

From Equation (2.2)αb = θ − ϕ

14

2.6. STEADY PROPELLER MODEL WITH OSCILLATORY FORCES

it follows∂ϕ

∂α= −∂αb

∂α(2.75)

and thus

∂ct∂α

= ∂ϕ

∂α

∂ct∂ϕ

= −∂ϕ∂α

∂ct∂αb

= −∂ϕ∂α

[∂cZ(k)∂αb

cos θ − ∂cX∂αb

sin θ]

(2.76)

∂cq∂α

= ∂ϕ

∂α

∂cq∂ϕ

= −∂ϕ∂α

∂cq∂αb

= −∂ϕ∂α

[∂cZ(k)∂αb

sin θ + ∂cX∂αb

cos θ]. (2.77)

Here, C(k) denotes Theodorsen’s function

C(k) = H(2)1 (k)

H(2)1 (k) + iH

(2)0 (k)

, (2.78)

a complex-valued combination of Hankel functions of the second kind. The reducedfrequency

k =2πf 1

2c

Vp=πc f

ωRVp

ωR

= πcfnλp

, (2.79)

is a measure of the unsteadiness of the flow and also referred to as Strouhal’snumber [3], where

fn = f

ωR(2.80)

is the normalized frequency of the airfoil vibration.

2.6 Steady propeller model with oscillatory forcesWhenever the propeller axis is inclined to the freestream, oscillations of the bladeinflow angle occur during every blade revolution, which results in a lag of theaerodynamic forces behind the motion. This phenomena can, as well, be handledby Theodorsen’s solution for thin airfoils oscillating in incompressible flow, whichhas already been introduced for the unsteady propeller model in Section 2.5. In thesteady-state case, however, not the instantaneous change of the aerodynamic forcesis relevant, but only the resulting phase lag. In accordance with Equation (2.74) thephase shift ∆γ can be determined as

∆γ = arg{1

2 ik + C(k)}. (2.81)

Here, the reduced frequency k depends on the rotational speed of the propeller ω,

k = ωc

2Vp=

ωcωR2Vp

ωR

= c

2λpR. (2.82)

15


The phase shift ∆γ can also be understood as displacement in angular position onthe propeller disk of the elemental forces and can be incorporated in the calculationof the force and moment coefficients as follows

dCFx = dCT dCMx = −dCQ

dCFy = dCQR

rsin(γ + ∆γ) dCMy = dCT

r

Rsin(γ + ∆γ) (2.83)

dCFz = dCQR

rcos(γ + ∆γ) dCMz = dCT

r

Rcos(γ + ∆γ).

16

Chapter 3

Implementation

Based on the above presented theory, a C++ software library for propeller analysishas been implemented.

The library basically consists of the following components:

• Blade Section Solver: Solution of the governing BEM equations and evaluationof section force and moment coefficients.

• Integrator: Adaptive integration of force and moment coefficients.

• Geometry and Aerodynamic Data: Representation of propeller geometry andaerodynamic data.

In the following sections further details on the implementation and the underlyingalgorithms will be presented. A detailed documentation of the library can be foundin Appendix D.

3.1 Solution of the governing BEM equations andevaluation of section force and moment coefficients

In order to solve the BEM equations (Equation (2.50) and (2.51)) an algorithmbased on Newton’s method is applied. The algorithm can be found in Algorithm1. A detailed derivation of the Jacobian matrix J used for the solution procedurecan be found in Appendix A. For this two-dimensional problem, the inverse ofthe Jacobian matrix can be directly determined. In order to improve the globalconvergence behavior, line search according to Armijo’s rule is applied to determinea step size ν, for which a sufficient decrease of ||F (x)|| is achieved [11] (lines 11-15):

||F (x+ νd)|| < (1− κν)||F (x)||. (3.1)

In the rare case that this procedure does not succeed with a reasonable step size, aminimum step size νmin is used.

17

CHAPTER 3. IMPLEMENTATION

Since the solution space contains a trivial solution, which will be found by theNewton approach, in some cases, when the initial value λa,initial is not appropriatelychosen, additional logic is necessary to detect and handle this case. The trivialsolution corresponds to the case

λa = −λxλr = λt

and can be easily detected by checking if

λ2p = (λx + λa)2 + (λt − λr)2 < tol. (3.2)

Selecting a suitable value for λa,initial is, however, not trivial and has to befurther discussed. One approach could be to compute an initial λa by settingλa = 0, λr = 0 and reduce f1 to a quadratic equation, which can be directly solvedfor λa. This procedure corresponds to approximating the thrust by means of theBlade Element Theory neglecting the induced velocities and then finding a λa, whichfulfills the momentum theory. Another approach could be to use an analyticalsolution of the BEM theory, as presented in [12], in order to have an initial guessfor λa. Both approaches, however, have turned out to be unsatisfactory in terms ofrobustness. Especially for the case when λx is so low that is has approximately thesame magnitude as λa and it holds λa,initial < 0, the initial guess might be so closeto the trivial solution that the solver will run into it.

Figure 3.1. Visualization of ϕ for different λa

This problem can be visualized by considering only the angle ϕ = tan−1 λx+λaλt

,where λr is neglected for simplicity, since it is generally so small that it does nothave much impact on ϕ. As shown in Figure 3.1 ϕλa=0 corresponds to the valuefor λa = 0 and ϕ0 = 0 to the trivial solution, where λx = −λa. In case that ϕini,which corresponds to a λa,initial provided by one of the above presented approaches,is lower than zero, and ϕsol, corresponding to the value for which the solution of theBEM equations is achieved, is greater than zero, it might happen that the solverwill end up in the trivial case rather than finding the right solution. Of course, thesame applies to the case ϕsol < 0, ϕini > 0. If, however, it holds ϕ0 < ϕsol < ϕini orϕini < ϕsol < ϕ0, the solver will probably find the right solution.

18

3.2. ADAPTIVE INTEGRATION OF FORCE AND MOMENT COEFFICIENTS

Based on these observations, a robust strategy for the selection of the λa,initialhas been developed. The initial λa will be chosen, so that a high value of ϕ, e.g.45◦, is achieved:

ϕ = tan−1 λx + λaλt

= π

4

⇔ tan π4 = 1 = λx + λaλt

⇒λa,initial = λt − λx. (3.3)

If the solver runs into the trivial solution, it is restarted using λa,initial = −λa,initial(lines 18-22). In order to increase the probability of starting with the right sign, ct(see Equation (2.61)) is evaluated for ϕλa=0. In most cases ϕλa=0 is close to ϕsoland, considering Equation (2.50), ct has always the same sign as λa.

In order to handle rare cases, for which no convergence can be achieved, as itmight happen when the solver is stuck in a local minimum, a first attempt is torestart the solver with λa,initial = −λa,initial, as well (lines 24-33).

3.2 Adaptive integration of force and moment coefficientsIn order to perform efficient numerical integration of the force and moment coefficientsover the propeller disk, an adaptive algorithm based on Newton-Cotes quadraturerules, as presented in [16], is used.

In order to integrate the function f in the interval [a, b], the interval is subdividedin a number of smaller intervals with the nodes t0, t1, t2, · · · , tn, where t0 = a andtn = b. The numerical integral value Ih is the sum of the subintegrals Ih,i,

Ih =n−1∑i=0

Ih,i, (3.4)

where each subintegral is approximated by a quadrature rule like the Simpson rule

Ih,i = ti+1 − ti6 [f(ti) + 4f(tm,i) + f(ti+1)] , (3.5)

withtm,i = ti+1 + ti

2 . (3.6)

The idea of the adaptive algorithm is to refine each subintegral by further dividingit in smaller intervals until the integration error ei is lower than a given tolerance:

ei = |Ih,i − Ii| < tol, (3.7)

where Ii is the exact value of the subintegral. This can be efficiently done by dividingthe integration interval of Ih,i in two subintervals of equal size, Ih,i,1 and Ih,i,2, sincein this case all three function values f(ti),f(tm,i) and f(ti+1) can be reused. Since

19


Algorithm 1 Solution of BEM equations1: x := (λa, λr)T2: F (x) := (f1(x), f2(x))T3: κ = 1e−4 (default value in [11])4: λa,initial = sgn(ct)(λt − λx)5: λr,initial = 0.0016: i = 07:8: while ||F (x)|| > ε do9: Compute Jacobian matrix J(x)

10: Compute search direction d = −J(x)−1F (x)11:12: ν = 113: while ||F (x+ νd)|| > (1− κν)||F (x)|| do14: ν = ν

215: end while16: x = x+ νd17:18: if λ2

p < tol and not restarted then19: λa = −λa,initial20: λr = λr,initial21: restart22: end if23:24: i = i+ 125: if i > imax then26: if not restarted then27: λa = −λa,initial28: λr = λr,initial29: restart30: else31: exit with error message32: end if33: end if34: end while

20

3.3. REPRESENTATION OF GEOMETRY AND AERODYNAMIC DATA

the exact integral value Ii is unknown, the integration error ei is approximated by(extrapolation principle, see [16])

ei ≈|(Ih,i,1 + Ih,i,2)− Ih,i|

2p − 1 , (3.8)

or more specifically for the Simpson rule (p=5)

ei ≈115 |(Ih,i,1 + Ih,i,2)− Ih,i|. (3.9)

An algorithm based on this concept is presented in Algorithm 2. The algorithmsubdivides the integration interval in two subintervals and then recursively refineseach of these subintervals until the error ei is small enough to fulfill the abortioncriterion.

The algorithm is applied to both, integration of the force and moment coefficients(see Equation (2.40) and (2.40)) and integration of the complex and frequency-dependent derivatives of the force and moment coefficients (see Equation (2.70)and (2.71)), in the same way. When integrating over the angular blade positionγ evaluation of the function value f(γ) corresponds to integration over the bladeradius for fixed γ:

I =∫ 2π

0f(γ)dγ, (3.10)

withf(γ) =

∫ R

RH

f(r, γ)dr. (3.11)

3.3 Representation of geometry and aerodynamic data

Figure 3.2. Airfoil sections at different radii.

The geometry of a propeller blade is basically defined by the geometry of theairfoil sections and the twist angle τ with which they are arranged (see Figure 3.2).

21


Algorithm 2 Adaptive Simpson Rule (adapted from [16]).// set initial valuesI = 0t1 = a, f1 = f(a)tm = a+b

2 , fm = f(tm)t2 = b, f2 = f(b)h = b− aS = h

6 (f1 + 4fm + f2)

function integral(t1, tm, t2, f1, fm, f2, S, h)// compute midpoints of left and right subinterval and function valuestl = 1

2(t1 + tm), fl = f(tl)tr = 1

2(tm + t2), fr = f(tr)h = h

2

// determine integral value of left and right subintegralSl = h

6 (f1 + 4fl + fm)Sr = h

6 (fm + 4fr + f2)SF = Sl + Sr

if 115 |SF − S| < tol(|SF |+ tol) thenI = I + SFreturn

else// refine left and right subintegral recursivelyintegral(t1, tl, tm, f1, fl, fm, Sl, h)integral(tm, tr, t2, fm, fr, f2, Sr, h)

end ifend function

In order to solve the BEM equations, it is, however not necessary to representthe actual geometry of the airfoil sections, since only the chord length c and theaerodynamic data of the airfoil, described by cL and cD-curves, are of relevance forthe calculations.

The distribution of chord length and blade twist over the blade radius is repre-sented as a simple table. An example can be found in Listing 3.1. The first columnof the chord and twist table corresponds to the non-dimensional radius

ndr = r −RR−RHub

. (3.12)

In order to determine values in between two data points, spline interpolation is used.The representation of the aerodynamic data is handled in a similar way. The

cL and cD-data of a specific airfoil geometry are stored as tabulated values in a

22

3.3. REPRESENTATION OF GEOMETRY AND AERODYNAMIC DATA

Listing 3.1. Example for the file format used to describe the propeller geometry<PropGeometry bladeCount ="4" hubRadius ="0.316" t ipRadius ="1.625" >

<ChordTable>0 . 0 0 .2 00 . 2 0 .2 10 . 4 0 .2 10 . 6 0 .2 00 . 8 0 .1 60 . 9 0 .1 01 . 0 0 .0 3

</ChordTable><TwistTable>

0.00000 31.361720.05000 28.359110.10000 25.521690.15000 22.859040.20000 20.375500.25000 18.071100.30000 15.942410.40000 12.186600.50000 9.043080.60000 6.436560.70000 4.292620.80000 2.543330.90000 1.129331.00000 0.00000

</TwistTable></PropGeometry>

<A i r f o i l S e t ><CoefTable ndr ="0.0" f i leName="hs1620x . dat"/><CoefTable ndr ="0.2" f i leName="hs1712x . dat"/><CoefTable ndr ="0.45" f i leName="hs1708x . dat"/><CoefTable ndr ="0.7" f i leName="hs1606x . dat"/><CoefTable ndr ="1.0" f i leName="hs1404x . dat"/>

</ A i r f o i l S e t >

simple text file. Listing 3.1 (see AirfoilSet) shows how the distribution of the airfoilsections is described by defining the non-dimensional radius and the correspondingfile containing the aerodynamic data for a number of sections. In order to providevalues for cL and cD over the whole blade radius and range of inflow angles αb splineinterpolation is used here as well.

23

Chapter 4

Validation

In order to carry out a validation of the BEM model, results obtained with themodel have been compared to experiments performed by the National AdvisoryCommittee for Aeronautics (NACA).

4.1 Parallel inflow-conditionsThe first set of experimental data is extracted from NACA Report 594 [15], wherethe performance of six propellers is measured over the full speed range at differentblade pitch angles. Although full-size propellers have been used, the wind tunnelspeed did not exceed 100 miles per hour and so, even for high advance ratios, theinflow velocity at the blade tips was with around 78ms far below the speed of sound.

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

Advance Ratio J

Thr

ust C

oeffi

cien

t CT

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Advance Ratio J

Pow

er C

oeffi

cien

t CP

BEMMeasured

BEMMeasured

Figure 4.1. Thrust and power coefficient for a blade angle θ of 35◦ at 0.75R.

25

CHAPTER 4. VALIDATION

Therefore, the results should not strongly be affected by compressibility effects. Forthe validation, the performance characteristic of Propeller Bx is used, since it reachedthe highest efficiency among all six tested propellers and, with a low twist anglenear the blade root, has a geometry, which is most similar to modern propellers.Propeller Bx has a diameter of 10.06 ft and has three blades composed of Clark Yairfoil sections. The description of the propeller geometry is provided in the reportand the aerodynamic data of the airfoil sections was generated with the softwareXFOIL [4].

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.05

0

0.05

0.1

0.15

0.2

Thru

stC

oeffi

cien

tCT

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

Sec

tion

Thru

stC

oeffi

cien

tCT’

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

10

20

30

Advance Ratio J

Sec

tion

Inflo

wA

ngle

α [d

eg

]b

BEMMeasured

BEM Section at 0.75R

BEM Section at 0.75R

αstall

Figure 4.2. The upper curve shows the thrust coefficient of the propeller. Themiddle and lower curve shows the section thrust coefficient and the inflow angle ofthe blade section at 0.75R. The vertical line indicates an advance ratio of 0.68, forwhich the stall point of the 0.75R section is reached. All three curves correspond to ablade angle of 35◦ at 0.75R.

Figure 4.1 shows the measured performance of the propeller in comparison tothe results of the BEM model. For the representation of the thrust coefficient CT

26

4.1. PARALLEL INFLOW-CONDITIONS

−80 −60 −40 −20 0 20 40 60 80−1.5

−1

−0.5

0

0.5

1

1.5

Inflow Angle α [deg] b

Lift

Coe

ffici

entc

L

Figure 4.3. Lift coefficient cL over inflow angle αB at the 0.75R blade section.

and power coefficient CP the definition by NACA is used

CT = T

ρn2D4 (4.1)

CP = P

ρn3D5 = 2πnQρn3D5 , (4.2)

where ρ is the fluid density, n the number of revolutions per second and D = 2R thepropeller diameter. The blade section coefficients are defined as

C ′j = dCjdx

, (4.3)

wherex = r

R. (4.4)

It can be seen that the BEM model severely deviates from the experimental resultsfor advance ratios lower than one. In this region the airfoil sections experience aninflow angle αB beyond the stall point over a large part of the blade. Figure 4.2shows the development of αB for the blade section at 0.75R. In this region of theblade the highest aerodynamic loads can usually be observed and therefore thesection is representative for the performance of the whole propeller. It can be seenthat the airfoil stalls for an advance ratio lower than 0.68. The low section thrustcoefficient at low advance ratios corresponds to the rapidly decreasing lift coefficientcL for inflow angles exceeding the stall point (compare Figure 4.3).

The software used for the generation of the aerodynamic data, XFOIL, makes useof a zonal approach, where integral boundary layer theory is coupled with an inviscidsolution of the flow field in regions away from the airfoil surface [4, 5]. The methodprovides good results as long as only small parts of the airfoil surface experienceseparated flow. In contrast, when flow separation occurs at a very early position onthe upper airfoil surface, as it is the case for high inflow angles, XFOIL is not able topredict the lift coefficient very well. In terms of robustness the BEM solver requiresthe aerodynamic data to be defined for an inflow angle αb ranging from −90◦ to 90◦.XFOIL, however, usually fails to provide results for inflow angles far beyond the

27


0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.2 0.4 0.6 0.8 1−10

−5

0

5

10

Blade Radius r/R0.2 0.4 0.6 0.8 120

25

30

35

40

45

Blade Radius r/R

Inflowangle

α [d

eg

]b

CT’

CQ’

CT’

CQ’

Figure 4.4. Section thrust coefficient, section torque coefficient and inflow angle atJ = 0 (left) and J = 1.2 (right) at a blade angle θ of 35◦ at 0.75R.

stall point. The aerodynamic data for regions, where XFOIL did not provide results,have been approximated by flat plate theory (see, e.g., [1]). It is assumed that inreality the lift coefficient in the post-stall region is higher than predicted by XFOILor flat plate theory. This would explain the large deviation of the BEM model fromthe experimental data at low advance ratios.

In order to further examine this behavior, in Figure 4.4 the inflow angle over theblade radius has been visualized for zero advance, where the model severely deviatesfrom the experimental data, and an advance ratio of 1.2, which is inside a regionwhere the model covers the performance characteristic of the propeller quite well.For zero advance, the inflow angles range in a region, where the lift coefficient isassumed to be poorly approximated, over the whole blade radius. In contrast, foran advance ratio of 1.2 the inflow angle ranges in the attached flow regime of thecL-curve, where XFOIL is expected to provide good accuracy.

Figure 4.5 shows the performance of the propeller for a blade angle of 15◦ at0.75R, which would be a more reasonable choice for low advance ratios. Here theoverall characteristic of the thrust and power curves is covered quite well by theBEM model. Again, this can be explained by considering the distribution of theinflow angle over the blade radius (see Figure 4.6). It can be seen that for bothcases the inflow angle ranges inside the attached flow regime of the cL-curve, wheregood accuracy of the aerodynamic data can be assumed.

The fact that the BEM model underpredicts the thrust for higher advance ratios(see Figures 4.1 and 4.5) is rather unexpected. Since the Blade Element MomentumTheory is based on a simplified and idealized flow field around the blade sections,

28

4.1. PARALLEL INFLOW-CONDITIONS

0 0.2 0.4 0.6 0.80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Advance Ratio J

Thr

ust C

oeffi

cien

t CT

0 0.2 0.4 0.6 0.80

0.01

0.02

0.03

0.04

0.05

Advance Ratio J

Pow

er C

oeffi

cien

t CP

BEMMeasured

BEMMeasured

Figure 4.5. Thrust and power coefficient for a blade angle θ of 15◦ at 0.75R.

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.2 0.4 0.6 0.8 10

5

10

15

20

Blade Radius r/R

Inflo

w A

ngle

αb [d

eg]

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

Blade Radius r/R

CT’

CQ

’

CT’

CQ

’

Figure 4.6. Section thrust coefficient, section torque coefficient and inflow angle atJ = 0 (left) and J = 0.6 (right) at a blade angle θ of 15◦ at 0.75R.

29


neglecting any three-dimensional effects, it is, in fact, expected to provide an upperlimit for the thrust coefficient. One explanation could be that the presented thrustmeasurements might be too high. This problem was discussed by the authors ofthe report, who studied the influence of the shape of the engine cowling on themeasurements. The test arrangement is shown in Figure 4.7. They observed that thepresence of the propeller slipstream resulted in a reduction of the drag force actingon the nacelle unit. The propeller thrust T = R+D was determined by measuringthe axial force R of the whole propeller-nacelle-unit and then adding the drag forceD for the corresponding air speed of the nacelle, measured with the propeller off. Ifthe actual value for the drag force of the nacelle unit, however, is reduced due tothe slipstream, this approach would result in too high values for the thrust.

Figure 4.7. Test arrangement and cowling shapes used for the measurements(reprinted from [15]).

4.2 Non-parallel inflow-conditionsIn NACA report 1205 [10] the effect of inclination of the propeller axis to theairstream on the thrust coefficient was observed. The thrust has been measured foran inclination angle α of 0◦, −4.55◦ and −9.8 at different blade sections for a numberof angular blade positions γ. A 10 ft propeller with three NACA 10-(3)(08)-03 bladeshaving NACA 16-series blade sections was used in the study. The blade geometry isprovided in the report and the airfoil data was generated by XFOIL. The tests havebeen performed at different rotational speeds ranging from 1350 rpm to 2160 rpm.For the validation, however, only data obtained at the lowest rotational speed of1350 rpm is considered to avoid compressibility effects as far as possible.

Figure 4.8 shows the development of the thrust coefficient with advance ratio andthe corresponding inflow angle at the blade section. As before, it can be seen that

30

4.2. NON-PARALLEL INFLOW-CONDITIONS

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.04

0.08

0.12

0.16

0.2

Sec

tion

Thr

ust C

oeffi

cien

t CT’

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4−5

0

5

10

15

Advance Ratio J

Inflo

w A

ngle

αb [d

eg]

BEMMeasured

αstall

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Sec

tion

Thr

ust C

oeffi

cien

t CT’

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2

3

Blade Radius r/R

Inflo

w A

ngle

αb

BEMMeasured

BEM

Figure 4.8. Left: Section coefficient and inflow angle of the 0.6R blade section overadvance ratio. Right: Section coefficient and inflow angle over blade radius at anadvance ratio of 1.2. Both curves correspond to an inclination angle of 0◦.

−90 −75 −60 −45 −30 −15 0 15 30 45 60 75 90

−1

−0.5

0

0.5

1

Inflow Angle αb

Lift

Coe

ffici

ent c

L

Figure 4.9. Lift coefficient cL over inflow angle αB at the 0.60R blade section.

the results of the BEM model are in good agreement with the experimental data aslong as the inflow angle does not exceed the stall point (see Figure 4.9). No data forthe overall propeller performance is available in the report, but measurements forblade sections at different radii are presented. According to Figure 4.10 the section at0.6R was chosen for the validation, since it showed less dependence on the rotationalspeed than sections at higher radii, which are severely affected by compressibilityeffects due to the higher tangential velocity. The tangential velocity of the 0.6Rsection at 1350 rpm is around 129ms . On the other hand, the section is close enoughto the region where the highest aerodynamic load occurs (see Figure 4.8) and thusshould reflect the overall performance of the propeller quite well.

Figure 4.11 and 4.12 show the change in section thrust coefficient with respectto the value at zero inclination over the angular blade position. The definition of

31


Figure 4.10. Effect of rotational speed on change in section thrust coefficient for aninclination angle of −4.55◦ (reprinted from [10]).

the angular blade position refers to the convention used in the report, where 0◦ is indirection of axis inclination, which, in this case, corresponds to the bottom center ofthe propeller disk. Not considering the oscillatory effects in the BEM model, thepeaks occur at the 90◦ and 270◦ position, as it would be expected by the nature of themodel. Incorporating the phase shift resulting from oscillations of the inflow angleat the blade section in the BEM model (see Section 2.6), a phase lag of 8.39◦ canbe determined. The phase lag of the experimental values, however, is much higherin magnitude. One reason for the discrepancy is probably that the effects of bladedeformation are not included in the model, which would contribute to the phase lag.On the other hand, the authors of the report attribute some amount of the phaselag of the experimental data to be caused by the measurement equipment. Thisespecially accounts for the difference in phase between the minimum and maximumpeak at an inclination angle of −4.55◦, for which they had no other explanation.

32

4.2. NON-PARALLEL INFLOW-CONDITIONS

0 30 60 90 120 150 180 210 240 270 300 330 360−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Angular Blade Position γ [deg]

Cha

nge

in S

ectio

n T

hrus

t Coe

ffici

ent ∆

CT’

BEMBEM oscil.Measured

Figure 4.11. Change in section thrust coefficient of the 0.6R blade section atan advance ratio of 1.2 over the angular blade position γ at an inclination angleα = −4.55◦. The horizontal error bars indicate a measurement accuracy of ±2◦.

0 30 60 90 120 150 180 210 240 270 300 330 360−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Angular Blade Position γ [deg]

Cha

nge

in S

ectio

n T

hrus

t Coe

ffici

ent ∆

CT’

BEMBEM oscil.Measured

Figure 4.12. Change in section thrust coefficient of the 0.6R blade section at anadvance ratio of 1.2 over the angular blade position γ at an inclination angle α = −9.8◦.The horizontal error bars indicate a measurement accuracy of ±2◦.

33

Chapter 5

Conclusion

An aerodynamic propeller model for load analysis and the implementation in asoftware library has been presented.

Validation of the steady-state model showed agreement with experimental datafor parallel and non-parallel inflow-conditions as long as the aerodynamic data beingused provided good accuracy.

The question whether the accuracy of the model is sufficient for structural loadanalysis could be a topic for future work. The accuracy of the model can potentiallybe improved by using a compressible formulation of the Momentum Theory. Forpractical application this would, however, mean that the aerodynamic data must beprovided for a range of Mach numbers.

35

Bibliography

[1] John D. Anderson. Fundamentals of Aerodynamics. McGraw-Hill, 2011.

[2] R.L. Bisplinghoff, H. Ashley, and R.L. Halfman. Aeroelasticity. Dover Pubns,1996.

[3] D. Borglund and D. Eller. Aeroelasticity of slender wing structures in low-speedairflow. Lecture Notes, KTH Stockholm, 2006.

[4] M. Drela. Xfoil: An analysis and design system for low reynolds number airfoils.Conference on Low Reynolds Number Airfoil Aerodynamics, University of NotreDame, 1989.

[5] M. Drela and M. Giles. Viscous-inviscid analysis of transonic and low reynoldsnumber airfoils. AIAA journal, 25(10):1347–1355, 1987.

[6] S. Drzewiecki. Théorie générale de l’hélice: hélices aériennes et hélices marines.Gauthier-Villars et cie., 1920.

[7] W. Froude. On the elementary relation between pitch, slip, and propulsiveefficiency. Transactions of the Institute of Naval Architects, 19:47–57, 1878.

[8] H. Glauert. Airplane propellers. In W.F. Durand, editor, Aerodynamic Theory,volume IV. Julius Springer, 1935.

[9] S. Goldstein. On the vortex theory of screw propellers. Proceedings of the RoyalSociety of London. Series A, 123(792):440–465, 1929.

[10] WH Gray, JM Hallissy, AR Heath, and NACA. Langley Research Center. Awind-tunnel investigation of the effects of thrust-axis inclination on propellerfirst-order vibration. 1954.

[11] C.T. Kelley. Solving nonlinear equations with Newton’s method, volume 1.Society for Industrial Mathematics, 2003.

[12] Robert S. Merrill. Nonlinear aerodynamic corrections to blade element momen-tum modul with validation experiments. Master’s thesis, Utah State University,2011.

37

BIBLIOGRAPHY

[13] L. Prandtl. Appendix to: A. betz, schraubenpropeller mit geringstem energiev-erlust. In Vier Abhandlungen zur Hydrodynamik und Aerodynamik. Kaiser-Wilhelm-Institut f. Strömungsforschung, 1927.

[14] W.J. Macquorn Rankine. On the mechanical principles of the action of propellers.Transactions of the Institute of Naval Architects, 6:13–39, 1865.

[15] T. Theodorsen, G.W. Stickle, M.J. Brevoort, and NACA. Langley ResearchCenter. Characteristics of six propellers including the high-speed range. 1937.

[16] W. Vogt. Adaptive Verfahren zur numerischen Quadratur und Kubatur. 2006.

[17] Q.R. Wald. The aerodynamics of propellers. Progress in Aerospace Sciences,42(2):85–128, 2006.

[18] F.E. Weick. Aircraft propeller design. McGraw-Hill Book Company, inc., 1930.

38

Appendix A

Jacobian matrix for the Newton solver

Derivation of the governing non-dimensional BEM equations

f1 = 12σλ

2pct − 2|λx + λa|λaF

={

12σλ

2pct − 2(λx + λa)λaF if λx + λa ≥ 0

12σλ

2pct + 2(λx + λa)λaF else

f2 = 12σλ

2pcq − 2|λx + λa|λrF

={

12σλ

2pcq − 2(λx + λa)λrF for λx + λa ≥ 0

12σλ

2pcq + 2(λx + λa)λrF else,

where

ct = cZ(αb) cos θ − cX(αb) sin θcq = cZ(αb) sin θ + cX(αb) cos θcZ = cL(αb) cosαb + cD(αb) sinαbcX = −cL(αb) sinαb + cD(αb) cosαb,

with respect to λa and λr yields

∂f1∂λa

=

12σλ

2p∂ct∂λa

+ 12σct

∂λ2p

∂λa− 2λaF − 2(λx + λa)F − 2(λx + λa)λa ∂F∂λa

if λx + λa ≥ 012σλ

2p∂ct∂λa

+ 12σct

∂λ2p

∂λa+ 2λaF + 2(λx + λa)F + 2(λx + λa)λa ∂F∂λa

else

∂f1∂λr

=

12σλ

2p∂ct∂λr

+ 12σct

∂λ2p

∂λr− 2(λx + λr)λa ∂F∂λr


2p∂ct∂λr

+ 12σct

∂λ2p

∂λr+ 2(λx + λr)λa ∂F∂λr

else

∂f2∂λa

=

12σλ

2p∂cq

∂λa+ 1

2σcq∂λ2

p

∂λa− 2λrF − 2(λx + λa)λr ∂F∂λa


2p∂cq

∂λa+ 1

2σcq∂λ2

p

∂λa+ 2λrF + 2(λx + λa)λr ∂F∂λa

else

∂f2∂λr

=

12σλ

2p∂cq

∂λr+ 1

2σcq∂λ2

p

∂λr− 2(λx + λa)F − 2(λx + λa)λr ∂F∂λr


2p∂cq

∂λr+ 1

2σcq∂λ2

p

∂λr+ 2(λx + λa)F + 2(λx + λa)λr ∂F∂λr

else,

39

APPENDIX A. JACOBIAN MATRIX FOR THE NEWTON SOLVER

with

∂ϕ

∂λa= ∂

∂λa

(arctan λx + λa

λt − λr

)=[(

1 +(λx + λaλt − λr

)2)(λt − λr)

]−1

∂ϕ

∂λr= ∂

∂λa

(arctan λx + λa

λt − λr

)= λx + λa

λ2p

∂ct∂λa

= ∂αb∂λa

∂ct∂αb

= − ∂ϕ

∂λa

∂ct∂αb

= − ∂ϕ

∂λa

(∂cZ∂αb


sin θ)

∂ct∂λr

= ∂αb∂λr

∂ct∂αb

= − ∂ϕ∂λr

∂ct∂αb

= − ∂ϕ∂λr

(∂cZ∂αb


sin θ)

∂cq∂λa

= ∂αb∂λa

∂cq∂αb

= − ∂ϕ

∂λa

∂cq∂αb

= − ∂ϕ

∂λa

(∂cZ∂αb


cos θ)

∂cq∂λr

= ∂αb∂λr

∂cq∂αb

= − ∂ϕ∂λr

∂cq∂αb

= − ∂ϕ∂λr

(∂cZ∂αb


cos θ)

∂λ2p

∂λa= ∂

∂λa

[(λx + λa)2 + (λt − λr)2

]= 2(λx + λa)

∂λ2p

∂λr= ∂

∂λr

[(λx + λa)2 + (λt − λr)2

]= −2(λt − λr)

∂F

∂ϕ= Nb(R− r)e−f cosϕπr√

1− e−2f sin2 ϕ∂F

∂λa= − ∂ϕ

∂λa

∂F

∂ϕ

∂F

∂λr= − ∂ϕ

∂λr

∂F

∂ϕ.

The Jacobian matrix J used for the solution of the BEM equations is defined as

J(x) := J(λa, λr) =(

∂f1∂λa

∂f1∂λr

∂f2∂λa

∂f2∂λr

).

40

Appendix B

Linearization of the BEM equationswith respect to α

Derivation of the non-dimensional BEM equations with respect to α yields

∂f1∂α

= 12σλ

2p

∂ct∂α

+ 12σct

∂λ2p

∂α− 2λaF

∂λx∂α− 2λaF

∂λa∂α− 2(λx + λa)F

∂λa∂α− 2(λx + λa)λa

∂F

∂α

∂f2∂α

= 12σλ

2p

∂cq∂α

+ 12σcq

∂λ2p

∂α− 2λrF

∂λx∂α− 2λrF

∂λa∂α− 2(λx + λa)F

∂λr∂α− 2(λx + λa)λr

∂F

∂α,

with

∂λx∂α

= −λ cosβ sinα∂λt∂α

= −λ cosβ cos γ cosα

∂λ2p

∂α= 2(λx + λa)

∂λx∂α

+ 2(λx + λa)∂λa∂α

+ 2(λt − λr)∂λt∂α− 2(λt − λr)

∂λr∂α

∂ϕ

∂α= 1λ2p

[(λt − λr)

∂λx∂α

+ (λt − λr)∂λa∂α− (λx + λa)

∂λt∂α

+ (λx + λa)∂λr∂α

]∂ct∂α

= ∂ϕ

∂α

∂ct∂ϕ

= −∂ϕ∂α

∂ct∂αb

= −∂ϕ∂α

[∂cZ(k)∂αb


sin θ]

∂ct∂α

= ∂ϕ

∂α

∂cq∂ϕ

= −∂ϕ∂α

∂ct∂αb

= −∂ϕ∂α

[∂cZ(k)∂αb


cos θ]

∂F

∂α= ∂ϕ

∂α

∂F

∂ϕ= −∂ϕ

∂α

Nb(R− r)e−f cosϕπr√

1− e−2f sin2 ϕ

In the operation point it must hold

∂f1∂α

∣∣∣∣α0

= 0 ∂f2∂α

∣∣∣∣α0

= 0,

41

APPENDIX B. LINEARIZATION OF THE BEM EQUATIONS WITH RESPECT TO α

which results in the linear system of equations[−1

2σ(∂cZ(k)∂αb


sin θ)

(λt − λr) + σct(λx + λa)− 2λaF − 2(λx + λa)F

+2(λx + λa)(λt − λr)λaNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λa∂α

+[−1

2σ(∂cZ(k)∂αb


sin θ)

(λx + λa)− σct(λt − λr)

+2(λx + λa)2λaNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λr∂α

=

−[−1

2σ(∂cZ(k)∂αb


sin θ)

(λt − λr) + σct(λx + λa)− 2λaF


1− e−2f sin2 ϕ

]∂λx∂α

−[1

2σ(∂cZ(k)∂αb


sin θ)

(λx + λa) + σct(λt − λr)

−2(λx + λa)2λaNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λt∂α

and [−1

2σ(∂cZ(k)∂αb


cos θ)

(λt − λr) + σcq(λx + λa)− 2λrF

+2(λx + λa)(λt − λr)λrNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λa∂α

+[−1

2σ(∂cZ(k)∂αb


cos θ)

(λx + λa)− σcq(λt − λr)− 2(λx + λa)F

+2(λx + λa)2λrNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λr∂α

=

−[−1

2σ(∂cZ(k)∂αb


cos θ)



1− e−2f sin2 ϕ

]∂λx∂α

−[1

2σ(∂cZ(k)∂αb


cos θ)

(λx + λa) + σcq(λt − λr)

−2(λx + λa)2λrNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λt∂α

which can be solved for the unknowns ∂λa∂α and ∂λr

∂α .

42

Appendix C

Linearization of the BEM equationswith respect to β

Derivation of the non-dimensional BEM equations with respect to β yields

∂f1∂β

= 12σλ

2p

∂ct∂β

+ 12σct

∂λ2p

∂β− 2λaF

∂λx∂β− 2λaF

∂λa∂β− 2(λx + λa)F

∂λa∂β− 2(λx + λa)λa

∂F

∂β

∂f2∂β

= 12σλ

2p

∂cq∂β

+ 12σcq

∂λ2p

∂β− 2λrF

∂λx∂β− 2λrF

∂λa∂β− 2(λx + λa)F

∂λr∂β− 2(λx + λa)λr

∂F

∂β,

with

∂λx∂β

= −λ cosα sin β

∂λt∂β

= λ sin γ cosβ + λ sinα cos γ sin β

∂λ2p

∂β= 2(λx + λa)

∂λx∂β

+ 2(λx + λa)∂λa∂β

+ 2(λt − λr)∂λt∂β− 2(λt − λr)

∂λr∂β

∂ϕ

∂β= 1λ2p

[(λt − λr)

∂λx∂β

+ (λt − λr)∂λa∂β− (λx + λa)

∂λt∂β

+ (λx + λa)∂λr∂β

]∂ct∂β

= ∂ϕ

∂β

∂ct∂ϕ

= −∂ϕ∂β

∂ct∂αb

= −∂ϕ∂β

[∂cZ(k)∂αb


sin θ]

∂ct∂β

= ∂ϕ

∂β

∂cq∂ϕ

= −∂ϕ∂β

∂ct∂αb

= −∂ϕ∂β

[∂cZ(k)∂αb


cos θ]

∂F

∂β= ∂ϕ

∂β

∂F

∂ϕ= −∂ϕ

∂β

Nb(R− r)e−f cosϕπr√

1− e−2f sin2 ϕ

In the operation point it must hold

∂f1∂β

∣∣∣∣α0

= 0 ∂f2∂β

∣∣∣∣α0

= 0,

43

APPENDIX C. LINEARIZATION OF THE BEM EQUATIONS WITH RESPECT TO β

which results in the linear system of equations[−1

2σ(∂cZ(k)∂αb


sin θ)

(λt − λr) + σct(λx + λa)− 2λaF − 2(λx + λa)F


1− e−2f sin2 ϕ

]∂λa∂β

+[−1

2σ(∂cZ(k)∂αb


sin θ)

(λx + λa)− σct(λt − λr)

+2(λx + λa)2λaNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λr∂β

=

−[−1

2σ(∂cZ(k)∂αb


sin θ)

(λt − λr) + σct(λx + λa)− 2λaF


1− e−2f sin2 ϕ

]∂λx∂β

−[1

2σ(∂cZ(k)∂αb


sin θ)

(λx + λa) + σct(λt − λr)

−2(λx + λa)2λaNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λt∂β

and [−1

2σ(∂cZ(k)∂αb


cos θ)



1− e−2f sin2 ϕ

]∂λa∂β

+[−1

2σ(∂cZ(k)∂αb


cos θ)

(λx + λa)− σcq(λt − λr)− 2(λx + λa)F

+2(λx + λa)2λrNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λr∂β

=

−[−1

2σ(∂cZ(k)∂αb


cos θ)



1− e−2f sin2 ϕ

]∂λx∂β

−[1

2σ(∂cZ(k)∂αb


cos θ)

(λx + λa) + σcq(λt − λr)

−2(λx + λa)2λrNb(R− r)e−f cosϕλ2pπr√

1− e−2f sin2 ϕ

]∂λt∂β

which can be solved for the unknowns ∂λa∂β and ∂λr

∂β .

44

Appendix D

Code documentation

libprop1.0

Generated by Doxygen 1.7.6.1

Sun Jul 8 2012 11:27:01

45

CONTENTS 1

Contents

1 Class Index 1

1.1 Class List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Class Documentation 2

2.1 AdaptiveIntegrator Class Reference . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Detailed Description . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Member Function Documentation . . . . . . . . . . . . . . . . 4

2.2 AdaptiveIntegratorInfo Struct Reference . . . . . . . . . . . . . . . . . 5


2.3 AirfoilSet Class Reference . . . . . . . . . . . . . . . . . . . . . . . . . 6



2.4 AirfoilSet::CoefTable Struct Reference . . . . . . . . . . . . . . . . . . 8

2.5 Integrator Class Reference . . . . . . . . . . . . . . . . . . . . . . . . 9



2.6 NPoint Struct Reference . . . . . . . . . . . . . . . . . . . . . . . . . . 10


2.7 Propeller Class Reference . . . . . . . . . . . . . . . . . . . . . . . . 11



2.8 PropGeometry Class Reference . . . . . . . . . . . . . . . . . . . . . 13


2.9 ScalarSpline Class Reference . . . . . . . . . . . . . . . . . . . . . . . 14


2.10 Section Class Reference . . . . . . . . . . . . . . . . . . . . . . . . . 15



2.11 SectionValues Struct Reference . . . . . . . . . . . . . . . . . . . . . . 18


1 Class Index

Generated on Sun Jul 8 2012 11:27:01 for libprop by Doxygen

1.1 Class List 2

1.1 Class List

Here are the classes, structs, unions and interfaces with brief descriptions:

AdaptiveIntegratorAdaptive integration of real-valued and complex functions 2

AdaptiveIntegratorInfoInformation about integration performed by AdaptiveIntegrator 5

AirfoilSetAirfoil data for propeller blade 6

AirfoilSet::CoefTable 8

IntegratorIntegration over the propeller disk 9

NPointNon-dimensional operating point 10

PropellerPropeller analysis interface 11

PropGeometryContainer for propeller geometry 13

ScalarSplineSimple cubic spline in one variable 14

SectionBlade section solver 15

SectionValuesInformation about solution of the blade section 18

2 Class Documentation

2.1 AdaptiveIntegrator Class Reference

Adaptive integration of real-valued and complex functions.

#include <adaptiveintegrator.h>

Static Public Member Functions

• template<class Functor , typename Type >

static void integrate1D (Functor &f, Type &I, Real xstart, Real xend, Real hmin,Real tol, AdaptiveIntegratorInfo &info)


2.1 AdaptiveIntegrator Class Reference 3

One-dimensional integration of a scalar integrand.


static void integrate1D (Functor &f, Type &I, Real xstart, Real xend, Real hmin,Real tol)


• template<class Functor , uint N, typename Type >

static void integrateVec1D (Functor &f, SVector< N, Type > &I, Real xstart, Realxend, Real hmin, Real tol, bool check[ ], AdaptiveIntegratorInfo &info)

One-dimensional integration of a integrand vector.


static void integrateVec1D (Functor &f, SVector< N, Type > &I, Real xstart, Realxend, Real hmin, Real tol, bool check[ ])


Static Private Member Functions


static void integrateVec1Drec (Functor &f, const SVector< N, Type > &S, S-Vector< N, Type > &I, const SVector< N, Type > &f1, const SVector< N, Type> &fm, const SVector< N, Type > &f2, Real x1, Real xm, Real x2, Real hmin,Real tol, bool check[ ], AdaptiveIntegratorInfo &info)

recursive call of the vector integration


static void integrate1Drec (Functor &f, Type S, Type &I, Type f1, Type fm, Type f2,Real x1, Real xm, Real x2, Real hmin, Real tol, AdaptiveIntegratorInfo &info)

recursive call of the scalar integration

2.1.1 Detailed Description

Adaptive integration of real-valued and complex functions.

Adaptive integration of a real-valued or complex function f (x) in the real valued interval[xstart ,xend ] using Simpson’s quadrature rule. The integrand f (x) can be scalar or avector

I =∫ xend

xstart

f (x)dx

The algorithm subdivides the integration interval [xstart ,xend ] in [xstart ,xmid ] and[xmid ,xend ] and then recursively refines each of the subintervals until for each subin-terval the estimated integration error is below a given tolerance tol. In case that theintegrand is a vector, a boolean array check[] (of the same length as the vector) has tobe given, which specifies for each component of the vector, if the error tolerance mustbe fulfilled for this component. The integration integral will be refined, until the toleranceis fulfilled for all components, where check[] is set to true.

A detailed description of the algorithm is presented in Vogt, W., Adaptive -Verfahren zur numerischen Quadratur und Kubatur, 2006.

The integrand has to be provided as a functor, where the function value is returned byoperator(). For example:


2.1 AdaptiveIntegrator Class Reference 4

class myFunctor{public:

...Real operator() (Real x){

Real y = ...return y;

}...

};

In order to retrieve information about the number of function evaluations and the min-imum integration intervall that have been used for the integration, the class Adaptive-IntegratorInfo can be used.

2.1.2 Member Function Documentation

2.1.2.1 template<class Functor , typename Type > static void AdaptiveIntegrator-::integrate1D ( Functor & f, Type & I, Real xstart, Real xend, Real hmin, Real tol,AdaptiveIntegratorInfo & info ) [inline, static]


Parametersf functor for the integrand functionI integral value to be returned

xstart lower bound of integration intervalxend upper bound of integration intervalhmin mininum stepsize that is allowed

tol tolerance for the estimated integration errorinfo information about integration

2.1.2.2 template<class Functor , typename Type > static void AdaptiveIntegrator-::integrate1D ( Functor & f, Type & I, Real xstart, Real xend, Real hmin, Real tol )[inline, static]




tol tolerance for the estimated integration error


2.2 AdaptiveIntegratorInfo Struct Reference 5

2.1.2.3 template<class Functor , uint N, typename Type > static voidAdaptiveIntegrator::integrateVec1D ( Functor & f, SVector< N, Type > & I, Realxstart, Real xend, Real hmin, Real tol, bool check[ ], AdaptiveIntegratorInfo & info) [inline, static]




tol tolerance for the estimated integration errorcheck specifies for each component of the vector if the tolerance has to be

fulfilledinfo information about integration

2.1.2.4 template<class Functor , uint N, typename Type > static voidAdaptiveIntegrator::integrateVec1D ( Functor & f, SVector< N, Type > & I, Realxstart, Real xend, Real hmin, Real tol, bool check[ ] ) [inline, static]




tol tolerance for the estimated integration errorcheck specifies for each component of the vector if the tolerance has to be

fulfilled

The documentation for this class was generated from the following file:

• adaptiveintegrator.h

2.2 AdaptiveIntegratorInfo Struct Reference

Information about integration performed by AdaptiveIntegrator.

#include <adaptiveintegrator.h>

Public Attributes

• uint numEval


2.3 AirfoilSet Class Reference 6

number of function evaluations

• Real minStepSize

minimum stepsize


Information about integration performed by AdaptiveIntegrator.

An AdaptiveIntegratorInfo object can be used in combination with AdaptiveIntegrator inorder to retrieve information about the integration, e.g. the number of function evaluationor the minimum stepsize that has been used.

The documentation for this struct was generated from the following file:

• adaptiveintegrator.h

2.3 AirfoilSet Class Reference

Airfoil data for propeller blade.

#include <airfoilset.h>

Classes

• struct CoefTable

Public Member Functions

• AirfoilSet ()

empty set

• void eval (Real ndr, Real ap, Real &CL, Real &CD) const

Evaluate lift and drag at non-dimensional radius and alpha parameter.

• void evalB (Real ndr, Real ap, Real &Cz, Real &Cx) const

Evaluate normal and axial force coefficient at non-dimensional radius and alpha.

• void tangent (Real ndr, Real ap, Real CL[ ], Real CD[ ]) const

Determine value and first derivative with respect to alpha parameter.

• void tangentB (Real ndr, Real ap, Real CZ[ ], Real CX[ ]) const

determine value and first derivative of normal and axial force coefficient with respectto alpha parameter

• Complex czak (Real ndr, Real ap, Real k) const

frequency-dependent normal force derivative

• void cak (Real ndr, Real ap, Real k, Complex CZ[ ], Real CX[ ]) const

determine value and first derivative of frequency-dependant normal force coefficientand axial force coefficient with respect to alpha parameter

• XmlElement toXml (bool share=false) const


2.3 AirfoilSet Class Reference 7

create XML representation• void fromXml (const XmlElement &xe)

recover contents from XML representation

Private Types

• typedef std::vector< CoefTable > CoefTableArray

Private Member Functions

• void appendTable (const std::string &fileName, Real ndr, CoefTableArray &tables)const

add (alpha, CL, CD) table to interpolation dataset• void interpolateTables (const CoefTableArray &tables)

interpolate tabulated dataset

Private Attributes

• SplineBasis m_radialBasis

spline basis for evaluation i radial direction• SplineBasis m_alphaBasis

spline basis for interpolation in alpha• Matrix m_pcl

control grid for lift coefficient• Matrix m_pcd

control grid for drag coefficient


Airfoil data for propeller blade.

AirfoilSet contains polars, that is, CL(alpha) and CD(alpha) as a smooth interpolationover both radial and alpha directions. In order to generate and initial representation, anairfoil set may be represented by an XmlElement file which references plain text filescontaining listings of alpha, CL and CD. Such listings must always cover the range from-90 degree to +90 degree alpha. In the case that the available data does not coverthe full range, a matlab tool in the ’polar’ folder can be used to extend polar data usingflat-plate theory.

AirfoilSet uses both radial and alpha parameters instead of the actual radius and angleof attack values. The definition of these non-dimensional parameters is:

r =r− rH

rT − rH

and

α =απ+

12,

so that both parameters are defined over the interval [0,1].


2.4 AirfoilSet::CoefTable Struct Reference 8


2.3.2.1 void AirfoilSet::eval ( Real ndr, Real ap, Real & CL, Real & CD ) const[inline]

Evaluate lift and drag at non-dimensional radius and alpha parameter.

For ap values exceeding the valid range, e.g 0 <= ap <= 1, the values for CL and CDare extrapolated as follows:

CL(ap > 1.0) = CL(1.0)+CL,ap(1.0)(ap−1.0)CL(ap < 0.0) = CL(0.0)+CL,ap(0.0)ap

CD(ap > 1.0) = CD(1.0)+CD,ap(1.0)(ap−1.0)CD(ap < 0.0) = CD(0.0)+CD,ap(0.0)ap.

2.3.2.2 void AirfoilSet::tangent ( Real ndr, Real ap, Real CL[ ], Real CD[ ] ) const[inline]

Determine value and first derivative with respect to alpha parameter.

For ap values exceeding the valid range, e.g 0 <= ap <= 1, the values for CL, CD, CL,apand CD,ap are extrapolated as follows:

CL(ap > 1.0) = CL(1.0)+CL,ap(1.0)(ap−1.0)CL(ap < 0.0) = CL(0.0)+CL,ap(0.0)ap

CD(ap > 1.0) = CD(1.0)+CD,ap(1.0)(ap−1.0)CD(ap < 0.0) = CD(0.0)+CD,ap(0.0)ap

CL,ap(ap > 1.0) = CL,ap(1.0)CL,ap(ap < 0.0) = CL,ap(0.0)CD,ap(ap > 1.0) = CD,ap(1.0)CD,ap(ap < 0.0) = CD,ap(0.0).

The documentation for this class was generated from the following files:

• airfoilset.h• airfoilset.cpp

2.4 AirfoilSet::CoefTable Struct Reference

Public Attributes

• Vector alpha• Vector CL• Vector CD• Real ndr


2.5 Integrator Class Reference 9


• airfoilset.h

2.5 Integrator Class Reference

Integration over the propeller disk.

#include <integrator.h>

Static Public Member Functions

• static void solve (const AirfoilSet &afs, const PropGeometry &geo, const NPoint&opp, Real c[ ])

integrate force and moment coefficients over the propeller disk for steady inflow con-ditions

• static void uSolveA (const AirfoilSet &afs, const PropGeometry &geo, const N-Point &opp, Real freq, Complex c[ ])

integrate complex and frequency dependent derivatives of the force and moment co-efficient with respect to alpha over the propeller disk for unsteady inflow conditions

• static void uSolveB (const AirfoilSet &afs, const PropGeometry &geo, const N-Point &opp, Real freq, Complex c[ ])

integrate complex and frequency dependent derivatives of the force and moment co-efficient with respect to beta over the propeller disk for unsteady inflow conditions


Integration over the propeller disk.

Integration of the force and moment coefficients over the propeller disk for steady inflowconditions.

Integration of the complex and frequency dependent derivatives of the force and mo-ment coefficient with respect to alpha or beta over the propeller disk for unsteady inflowconditions.


2.5.2.1 void Integrator::solve ( const AirfoilSet & afs, const PropGeometry & geo,const NPoint & opp, Real c[ ] ) [static]

integrate force and moment coefficients over the propeller disk for steady inflow condi-tions

Parametersafs airfoil data

geo propeller geometryopp non-dimensional operation point

c force and moment coefficients to be returned (array of length six)Generated on Sun Jul 8 2012 11:27:01 for libprop by Doxygen

2.6 NPoint Struct Reference 10

2.5.2.2 void Integrator::uSolveA ( const AirfoilSet & afs, const PropGeometry & geo,const NPoint & opp, Real freq, Complex c[ ] ) [static]

integrate complex and frequency dependent derivatives of the force and moment coef-ficient with respect to alpha over the propeller disk for unsteady inflow conditions


geo propeller geometryopp non-dimensional operation pointfreq normalized frequency, which is defined as fn =

fωR .

c complex-valued derivatives of force and moment coefficients to be re-turned (array of length six)

2.5.2.3 void Integrator::uSolveB ( const AirfoilSet & afs, const PropGeometry & geo,const NPoint & opp, Real freq, Complex c[ ] ) [static]

integrate complex and frequency dependent derivatives of the force and moment coef-ficient with respect to beta over the propeller disk for unsteady inflow conditions


geo propeller geometryopp non-dimensional operation pointfreq normalized frequency, which is defined as fn =

fωR .

c complex-valued derivatives of force and moment coefficients to be re-turned (array of length six)


• integrator.h• integrator.cpp

2.6 NPoint Struct Reference

Non-dimensional operating point.

#include <npoint.h>


• NPoint ()

create default point


2.7 Propeller Class Reference 11

Public Attributes

• Real advanceRatio

non-dimensional advance ratio

• Real bpitchAngle

blade pitch angle

• Real alpha

vertical inflow angle

• Real beta

horizontal inflow angle


Non-dimensional operating point.

The non-dimensional operation point is defined by the non-dimensional rate of advanceλ = V∞

ωR , the pitch angle of the blades, the vertical inflow angle α and the horizontalinflow angle β . The angles have to be given in radians.

See also

Propeller


• npoint.h

2.7 Propeller Class Reference

Propeller analysis interface.

#include <propeller.h>


• Propeller ()

create empty propeller object

• Propeller (const PropGeometry &geo, const AirfoilSet &afdata)

construct from airfoil data and external geometry

• void rename (const std::string &s)

change propeller name

• const std::string & name () const

access propeller name

• void eval (const NPoint &op, Real c[ ]) const

determine force and moment coefficients for given operation point

• void sweepPitch (const NPoint &op, Matrix &mcf, int np=51) const


2.7 Propeller Class Reference 12

perform a sweep over blade pitch (debugging/validation)

• Real bestBladePitch (const NPoint &op) const

determine optimal blade pitch angle for given operation point

• void createTable (uint nJ, uint nalpha, Real Jmax, Real alphaMax, NDArray< 3,Real > &ctab) const

write a force interpolation table to be read into matlab

• void createTable (const Vector &Jv, const Vector &alfav, const Vector &phiv, ND-Array< 4, Real > &ctab) const

write a force interpolation table to be read into matlab


export propeller definition to XML

• void fromXml (const XmlElement &xe)

recover definition from XML

Private Attributes

• PropGeometry m_geo

propeller geometry, blade count etc.

• AirfoilSet m_airfoil

airfoils used along the blade radius

• std::string m_sid

propeller identification


Propeller analysis interface.

Top-level interface for propeller analysis. Objects of class Propeller own geometry andairfoil data provide an interface to evaluate force and moment coefficients for a givennon-dimensional operating point.

Force and moment coefficients are made non-dimensional using the following relations

CFx =2Fx

ρ(ωR)2πR2

CFy =2Fy

ρ(ωR)2πR2

CFy =2Fy

ρ(ωR)2πR2

CMx =2Mx

ρ(ωR)2πR3

CMy =2My

ρ(ωR)2πR3

CMz =2Mz

ρ(ωR)2πR3 .


2.8 PropGeometry Class Reference 13

See also

PropGeometry, AirfoilSet, NPoint


2.7.2.1 void Propeller::eval ( const NPoint & op, Real c[ ] ) const

determine force and moment coefficients for given operation point

Parametersop non-dimensional operation point

c array of length 6 to store force and moment coefficients to be returned


• propeller.h• propeller.cpp

2.8 PropGeometry Class Reference

Container for propeller geometry.

#include <propgeometry.h>


• PropGeometry ()

undefined propeller geometry

• Real solidity (Real ndr) const

compute blade solidity fraction at non-dimensional radius r

• Real bladeRadius (Real ndr) const

compute dimensional radius at non-dimensional coordinate ndr

• Real bladeChord (Real ndr) const

interpolate chord at non-dimensional radius coordinate ndr

• Real bladeTwist (Real ndr) const

interpolate twist

• Real bladeLength () const

blade length

• uint bladeNumber () const

number of blades


return XML representation


recover from XML representation


2.9 ScalarSpline Class Reference 14


• void tableToSpline (const XmlElement &xe, ScalarSpline &spl) const

convert text table embedded in XML description into spline

Private Attributes

• Real m_hubRadius

hub radius

• Real m_tipRadius

blade tip radius

• uint m_nblade

number of blades

• ScalarSpline m_chordSpline

blade chord distribution

• ScalarSpline m_twistSpline

blade twist distribution


Container for propeller geometry.

This class holds the external geometry definitions of the propeller and stores a splineinterpolation of the radial chord and twist distribution.


• propgeometry.h• propgeometry.cpp

2.9 ScalarSpline Class Reference

Simple cubic spline in one variable.

#include <scalarspline.h>


• ScalarSpline ()

undefined spline

• Real eval (Real t) const

evaluate spline

• Real derive (Real t, uint k=1) const

first derivative

• void scale (Real f)


2.10 Section Class Reference 15

modify control point values by scaling

• void interpolate (const Vector &x, const Vector &y)

create spline by interpolating a set of values


return XML representation


recover contents from XML representation

Private Attributes

• SplineBasis m_basis

basis

• Vector m_cp

control point values


Simple cubic spline in one variable.

A very simple one-variable spline object limited to cubic interpolation.


• scalarspline.h• scalarspline.cpp

2.10 Section Class Reference

Blade section solver.

#include <section.h>


• Section (const AirfoilSet &airfoilset, const PropGeometry &propgeometry, constNPoint &oppoint)

initialize section

• bool solve (const Real ndr, const Real gamma, Vct6 &c)

obtain force and moment coefficient for blade section at non-dimensional radius ndrand angular blade position γ

• bool solve (const Real ndr, const Real gamma, Vct6 &c, SectionValues &sval)


• bool dcoeffAlpha (const Real ndr, const Real gamma, const Real freq, CpxVct6&dc, Complex &dla, Complex &dlr)



Obtain frequency-dependent derivatives of elemental force and moment coefficientswith respect to alpha for given normalized frequency.

• bool dcoeffBeta (const Real ndr, const Real gamma, const Real freq, CpxVct6&dc, Complex &dla, Complex &dlr)

Obtain frequency-dependent derivatives of elemental force and moment coefficientswith respect to beta for given normalized frequency.


• Real TL (const Real phi, const Real r, const Real R) const

Prandtl tip loss factor.

• Real getInitial (Real ndr) const

get initial value for induced velocity

• void Jacobian (Real ndr, Real va, Real vr, Real f[ ], Real J[ ], const Real h) const

approximate Jacobian of F with central finite differences

• void F (Real ndr, Real va, Real vr, Real &f1, Real &f2) const

Evaluate values of the governing equations of the Blade Element Momentum equa-tions.

• void JacobianA (Real ndr, Real la, Real lr, Real f[ ], Real J[ ]) const

evaluate F and Analytical Jacobian of F

Private Attributes

• const AirfoilSet m_airfoilset

contains airfoil sections used along the propeller blade

• const PropGeometry m_propgeometry

geometry of the propeller blade

• const NPoint m_oppoint

operation point

• Real m_r

radius of the blade section

• Real m_R

tip radius

• Real m_theta

theta angle (sum of blade twist angle and blade pitch angle)

• Real m_stheta

sine of theta

• Real m_ctheta

cosine of theta

• Real m_J

advance ratio

• Real m_AoA

angle of attack with respect to propeller axis

• Real m_beta



sideslip angle with respect to propeller axis

• Real m_Jx

axial advance ratio

• Real m_Jt

tangential advance ratio

• Real m_sigma

blade solidity


Blade section solver.

1) Solution of the governing Blade Element Momentum equations in order to determinethe forces and moments (coefficients) that are acting on the blade section at a givenradius and angular position of the blade.

2) Obtain frequency-dependent derivatives of the force and moment coefficients withrespect to α or β for the blade section at a given radius, angular blade position andfrequency.


2.10.2.1 bool Section::dcoeffAlpha ( const Real ndr, const Real gamma, const Real freq,CpxVct6 & dc, Complex & dla, Complex & dlr )

Obtain frequency-dependent derivatives of elemental force and moment coefficientswith respect to alpha for given normalized frequency.

Parameters

ndr non-dimensional radius ndr = r−RHubRTip−RHub

gamma angular blade position γ [rad]freq normalized frequency fn =

fωR

dc complex-valued derivatives of force and moment coefficients to be re-turned

dla derivate of axial induced velocity ∂∂α λa to be returned

dla derivate of tangential induced velocity ∂∂α λr to be returned

2.10.2.2 bool Section::dcoeffBeta ( const Real ndr, const Real gamma, const Real freq,CpxVct6 & dc, Complex & dla, Complex & dlr )

Obtain frequency-dependent derivatives of elemental force and moment coefficientswith respect to beta for given normalized frequency.

Parameters


gamma angular blade position γ [rad]


2.11 SectionValues Struct Reference 18

freq normalized frequency fn =f

ωRdc complex-valued derivatives of force and moment coefficients to be re-

turneddla derivate of axial induced velocity ∂

∂β λa to be returned

dla derivate of tangential induced velocity ∂∂β λr to be returned

2.10.2.3 bool Section::solve ( const Real ndr, const Real gamma, Vct6 & c ) [inline]


Parameters


gamma angular blade position γ [rad]c force and moment coefficients to be returned

2.10.2.4 bool Section::solve ( const Real ndr, const Real gamma, Vct6 & c,SectionValues & sval )


Parameters


gamma angular blade position γ [rad]c force and moment coefficients to be returned

sval detailed information about the solution


• section.h• section.cpp

2.11 SectionValues Struct Reference

Information about solution of the blade section.

#include <section.h>

Public Attributes

• Real r

radius of current section


2.11 SectionValues Struct Reference 19

• Real R

tip radius

• Real l

advance ratio λ• Real lx

axial advance ratio λx

• Real lt

tangential advance ratio λt

• Real la

ratio of axial induction λa

• Real lr

ratio of tangential induction λa

• Real lp2

λ 2p

• Real phi

ϕ• Real alphaB

blade inflow angle αb

• Real theta

θ• Real sigma

blade solidity σ• Real ct

thrust component ct

• Real cq

torque component cq

• Real F

tip loss factor

• Real AoA

angle of attack α• Real beta

angle of attack β


Information about solution of the blade section.

Can be used in combination with Section in order to retrieve detailed information aboutthe solution.


• section.h


TRITA-MAT-E 2012:02

ISRN-KTH/MAT/E--12/02-SE

www.kth.se

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