Dynamics of a 2D aircraft wing with cavity r15Dynamics of a 2D aircraft wing with cavity iii Abstract Within the framework of the European project VortexCell 2050, the dynamical behavior

Dynamics of a 2D aircraft wing with cavity

Test of a new experimental method

Marlies van Osch

DCT 2008.035

Master’s thesis Coaches: Prof. Dr. H. Nijmeijer

Prof. Dr. A. Hirschberg Dr. Ir. I. Lopez Ir. W.F.J. Olsman

Supervision: Prof. Dr. H. Nijmeijer Committee: Prof. Dr. H. Nijmeijer

Prof. Dr. A. Hirschberg Dr. Ir. I. Lopez Dr. Ir. R.J.M. Bastiaans Ir. W.F.J. Olsman (advisor)

Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control group Eindhoven, March 2008

Dynamics of a 2D aircraft wing with cavity ii

Dynamics of a 2D aircraft wing with cavity iii

Abstract Within the framework of the European project VortexCell 2050, the dynamical behavior of an airfoil with cavity needs to be assessed. The aim of this cavity is to stabilize a vortex which should reduce flow separation. This thesis presents a preliminary research to study experimentally the impact of a cavity on the steady and oscillating dynamical behavior of an airfoil. To achieve this aim, a new experimental method is verified. The experiments are performed in the wind tunnel in which the flow oscillates while the airfoil remains fixed. The oscillations of the flow are generated by acoustic waves, which are produced by two speakers in the side walls of the wind tunnel. The experimental airfoil has a NACA 0018 profile. Measurement of a local pressure difference between the top- and bottom sides of the airfoil provides information on the dynamic response. The stationary behavior of the airfoil corresponds to the time averaged signal. The oscillating one is determined by a lock-in data analysis method which uses the driving signal to the amplifier of the speakers as reference. Measurements are carried out for various flow velocities, amplitudes of acoustic perturbation, and angles-of-attack. Firstly, a comparison is made between the NACA 0018 airfoil and a theoretical model, which is the thin airfoil model. The measurement method is proved suitable for determination of the static and oscillating dynamical behavior of the airfoil. Difference between the theoretical and experimental values can be explained by the finite thickness of the airfoil. Hereafter, in the NACA 0018 profile a cavity is made and the influence of this cavity on the dynamical behavior is investigated. The cavity clearly affects both the stationary and oscillating dynamic response of the airfoil.

Dynamics of a 2D aircraft wing with cavity iv

Dynamics of a 2D aircraft wing with cavity v

Samenvatting In het kader van het Europese project VortexCell 2050 is het noodzakelijk om het dynamisch gedrag van een vliegtuig vleugel met caviteit vast te stellen. Het nut van deze caviteit is een vortex in de holte te stabiliseren en zo loslating van de stroming over de vleugel te minimaliseren. Dit project beschrijft een experimenteel vooronderzoek van de impact van een caviteit op het dynamisch gedrag van een vleugel. Om dit te bereiken is een nieuwe experimentele methode geverifieerd. De experimenten zijn uitgevoerd in een windtunnel waar, in plaats van de vleugel te bewegen ten opzichte van de stroming, nu de luchtstroming in trilling is gebracht terwijl de vleugel vast wordt gehouden. De oscillaties in de stroming zijn gegenereerd door akoestische golven welke opgewekt zijn door twee luidsprekers in de zijwanden van de windtunnel. Er is gekozen voor een vleugel met een NACA 0018 profiel. Het locale drukverschil over de vleugel is een maat voor het dynamisch gedrag. Het stationaire gedrag is bepaald door een tijdsgemiddelde van het signaal. Het oscillerende gedrag is verkregen door een lock-in data methode toe te passen welke het ingangssignaal naar de versterker van de luidsprekers gebruikt als referentie signaal. De metingen zijn uitgevoerd voor verschillende luchtsnelheden, sterktes van akoestische golven en aanstroom hoeken. In het begin zijn de experimentele waardes van de NACA 0018 vleugel vergeleken met de waardes volgens een analytisch model. Dit model beschouwt the vleugel als een dunne vlakke plaat. Op deze manier is aangetoond dat de experimentele methode geschikt is om zowel de statische als oscillerende dynamisch gedrag van de vleugel te bepalen. Het verschil tussen de experimentele en analytische waardes kunnen verklaard worden door de eindige dikte van de experimentele vleugel. Vervolgens is aan de bovenkant van een NACA 0018 vleugel een caviteit gemaakt en zijn de experimenten herhaald om de invloed van de caviteit vast te stellen. Een caviteit veroorzaakt een duidelijk verschil in zowel het statische als het oscillerende dynamisch gedrag van de vleugel.

Dynamics of a 2D aircraft wing with cavity vi

Dynamics of a 2D aircraft wing with cavity vii

Contents Abstract .......................................................................................................................................................iii Samenvatting ...............................................................................................................................................v Contents .....................................................................................................................................................vii Nomenclature & abbreviations ........................................................................................................... xi 1 Introduction ..............................................................................................................................................1

1.1 Vortex cell...........................................................................................................................................1 1.2 Objective.............................................................................................................................................2 1.3 Outline.................................................................................................................................................3 1.4 Airfoil geometry.................................................................................................................................3

2 Thin airfoil model ...................................................................................................................................5

2.1 Specification problem .......................................................................................................................5 2.1.1 Assumptions .............................................................................................................................6 2.1.2 Boundary conditions................................................................................................................6

2.2 Mathematical techniques ..................................................................................................................7 2.2.1 Acceleration potential..............................................................................................................8 2.2.2 Conformal mapping.................................................................................................................8 2.2.3 Boundary conditions................................................................................................................9

2.3 Stationary airfoil.................................................................................................................................9 2.3.1 Complex acceleration potential............................................................................................10 2.3.2 Flow field.................................................................................................................................12 2.3.3 Stationary pressure difference ..............................................................................................13

2.4 Oscillating airfoil..............................................................................................................................14 2.4.1 Complex acceleration potential............................................................................................15 2.4.2 Flow field.................................................................................................................................16 2.4.3 Oscillating pressure difference.............................................................................................17

3 Experimental method..........................................................................................................................19

3.1 Experimental setup .........................................................................................................................19 3.1.1 Test section .............................................................................................................................20 3.1.2 Experimental wing .................................................................................................................22 3.1.3 Data acquisition......................................................................................................................24

3.2 Data analysis: the Lock-in method ...............................................................................................24 3.2.1 Basic principle.........................................................................................................................24 3.2.2 Practical application...............................................................................................................25 3.2.3 Error.........................................................................................................................................26

Dynamics of a 2D aircraft wing with cavity viii

4 Acoustics of the wind tunnel .............................................................................................................29

4.1 Theoretical background..................................................................................................................29 4.1.1 Basic equations .......................................................................................................................29 4.1.2 Pipe modes in wind tunnel ...................................................................................................31 4.1.3 Uniform velocity in wind tunnel..........................................................................................33 4.1.4 Acoustic velocity in wind tunnel with uniform flow........................................................37

4.2 Experiments .....................................................................................................................................38 4.2.1 Semi-anechoic room..............................................................................................................38

Empty test section ........................................................................................................................39 Effect of an airfoil ........................................................................................................................39

4.2.2 Wind tunnel.............................................................................................................................40 Empty test section ........................................................................................................................40 Effect of an airfoil ........................................................................................................................44 Effect airfoil and uniform velocity ............................................................................................47

4.3 Transversal acoustic velocity .........................................................................................................47 4.3.1 Experimental determination of transversal acoustic velocity .........................................48

Acoustic velocity at the airfoil ....................................................................................................48 Acoustic velocity according to the pressure transducers at x = -2.5b.................................49

4.3.2 Comparison transversal acoustic velocities........................................................................50 4.3.3 Conclusion ..............................................................................................................................51

5 Dynamic response of NACA 0018 airfoil .......................................................................................53

5.1 Experimental parameters ...............................................................................................................53 5.1.1 Dimensionless pressure difference......................................................................................53 5.1.2 Experimental variables ..........................................................................................................55

5.2 Comparison NACA 0018 airfoil to Thin airfoil model.............................................................56 5.2.1 Stationary local pressure coefficient....................................................................................56

Influence angle-of-attack.............................................................................................................56 5.2.2 Oscillating local pressure coefficient...................................................................................58

Influence Strouhal number .........................................................................................................58 Influence amplitude acoustic perturbation ..............................................................................60 Influence angle-of-attack.............................................................................................................60

5.3 Conclusion........................................................................................................................................63 6 Influence of a cavity .............................................................................................................................65

6.1 Experimental comparison airfoil with cavity and NACA 0018 ...............................................65 6.1.1 Stationary local pressure coefficient....................................................................................65

Influence angle-of-attack.............................................................................................................65 6.1.2 Oscillating local pressure coefficient...................................................................................66

Influence Strouhal number .........................................................................................................66 Signal in cavity...............................................................................................................................68 Influence amplitude acoustic perturbation ..............................................................................70 Influence angle-of-attack.............................................................................................................72

6.2 Conclusion........................................................................................................................................76 7 Conclusions and recommendations................................................................................................77

7.1 Conclusions ......................................................................................................................................77 7.2 Recommendations...........................................................................................................................79

Bibliography ..............................................................................................................................................81

Dynamics of a 2D aircraft wing with cavity ix

A Investigation of Airfoil self noise ....................................................................................................85 A.1 Airfoil self noise..............................................................................................................................85 A.2 Self noise of NACA 0018..............................................................................................................86 A.3 Self noise of airfoil with cavity .....................................................................................................89 A.4 Conclusion.......................................................................................................................................91

B Mathematical techniques ..................................................................................................................93

B.1 Acceleration potential ....................................................................................................................93 B.2 Conformal mapping .......................................................................................................................96

C Boundary conditions thin airfoil model adapted to complex ζ-plane .................................99 D Determination constants A and B oscillating thin airfoil .....................................................103

D.1 Determination constant B ..........................................................................................................103 D.2 Determination constant A ..........................................................................................................103

E Geometry NACA 0018.......................................................................................................................109 F Photos experimental setup .............................................................................................................. 111 G Lock-in method..................................................................................................................................113

G.1 Mathematical principles ..............................................................................................................113 G.2 Hilbert transform .........................................................................................................................115

H In-approximation piezo pressure transducer PCB .................................................................117 I Higher order acoustical modes in wind tunnel .........................................................................119 J Data signals...........................................................................................................................................123

J.1 Data signal NACA 0018 ...............................................................................................................123 J.2 Data signal airfoil with cavity .......................................................................................................125 J.3 Data signal in cavity .......................................................................................................................126

Dynamics of a 2D aircraft wing with cavity x

Dynamics of a 2D aircraft wing with cavity xi

Nomenclature & abbreviations

A constant a acceleration a1 even Fourier

coefficient B constant b half chord length b1 odd Fourier

coefficient C Theodorson’s

function c 1) chord length 2) speed of sound cp stationary local

pressure coefficient

)/(2

021 Upc p ρ∆=

pc oscillating local

pressure coefficient over airfoil,

)ˆ/(ˆˆ02

1 vUpc p ρ∆=

d 1) thickness airfoil 2) location pressure transducers on airfoil

F 1) integrated pressure force 2) Joukowski transformation function

f 1) frequency 2) response signal g output signal Lock-in

method H Hilbert transform HALE High Altitude Long

Endurance i imaginary number

(time), 1 2 −=i

j imaginary number

(position), 1 2 −=j

k wavenumber, λπω /2/ == ck

K0 zero order modified Bessel function of second kind

K1 first order modified Bessel function of second kind

L 1) width wind tunnel section

2) characteristic length M Mach number,

cUM /=

P dynamic pressure, 2

21 UP ρ=

p pressure r 1) polar coordinate

(radius) in ζ –plane 2) reference signal lock-in method

Re Reynolds number, υ/Re UL=

sgn sign-function St Strouhal number, UbSt / ω= t time T oscillating period U Main velocity u velocity in x-direction v velocity in y-direction w 1) complex

acceleration potential 2) span width airfoil

Dynamics of a 2D aircraft wing with cavity xii

x x-coordinate in z-plane

y 1) position airfoil 2) y-coordinate in z-plane

y0 dimensionless position

airfoil, byy /0 =

z z-plane, ),( yxzz =

α angle-of-attack

∆ difference

δ tolerance

ε distance between two transducers in the flange

ζ complex ζ-plane, ),( ηξζζ =

θ polar coordinate (angle) in ζ-plane

η η-coordinate in ζ-plane

λ 1) wavelength 2) value to indicate a point on the airfoil,

20 ≤≤ λ

ν kinematic viscosity

ξ ξ-coordinate in ζ-plane

ρ density

φ 1) acceleration potential

2) phase

ψ acceleration streamfunction

ω angular frequency, f 2πω =

* convolution operator Superscript ‘ oscillating ^ amplitude → vector -1 inversed * 90 degrees shifted + propagating in

positive direction - propagating in

negative direction Subscripts airfoil airfoil c cut-off exp experimental fluid fluid h homogeneous solution m m-th mode n n-th mode res resonance ref reference x x-direction y y-direction z z-direction 0 reference state 1 pressure dipole at

1−=ξ


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Chapter 1

Introduction

1.1 Vortex cell

HALE (High Altitude Long Endurance) aircraft has to stay for long period at high altitude. As the air density on these heights is reduced, the ratio between the length and wing surface should be large (slender wings) to generate enough lift force. From structural viewpoint, long wings should be thick. The flow is unable to follow thick wing shapes and will separate from the airfoil. Consequently, the aerodynamical lift force is reduced and additional unsteady drag forces are generated. To overcome this problem a new design for the airfoil is made. The basic idea is to trap the vortices of the wake in a cavity, which is located at the top of the airfoil. These vortices will pull the flow over the airfoil and the flow becomes smooth again. A picture of such an airfoil is given in figure 1.

(a) (b)

Figure 1 (a) Thick airfoil with vortex shedding, (b) thick airfoil with a trapped vortex [1]

The idea of trapping vortices is not new. A Kasper wing is based on this vortex trapping. The wing consists of a plate with “spikes”. At the points of these spikes vortices are generated and subsequently trapped in the cavity between two adjacent spikes. The flow over the Kasper wing experiences again a smooth profile. (See figure 2).

Figure 2 Kasper wing [1]

This wing was the first which successfully applied the trapped vortices in flight. However, later wind tunnel experiments gave disappointing characteristics and led to the conclusion that this


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wing shouldn’t be able to fly. Vibrations in flight may have stabilized the vortices. In the framework of the European project VortexCell 2050 it will be investigated whether an airfoil with one cavity, in which the vortex will be stabilized, is a better design. This project will combine two technologies, trapped-vortex and active flow control. Eindhoven University of Technology has joined this project and is responsible for the study of the dynamical behavior of the airfoil. The risks related to possible vibrations have to be investigated. This includes determination of the oscillating forces due to vortices shedding or as a response to a fluctuating main flow. The dynamical behavior is modeled by a 2D numerical calculation and needs to be verified by experiments.

1.2 Objective

The object in this thesis is to study experimentally the steady and oscillating dynamical behavior of an airfoil with cavity. For this, a new experimental method needs to be verified to obtain the dynamical behaviors. The objective is divided into the following aspects

Analysis of the stationary and oscillating dynamical behavior of an thin airfoil model

Validate the measurement method by comparison experimental

behavior and the dynamical behavior according to the thin airfoil model

Study the impact of a cavity on the stationary and oscillating

dynamical behavior of an airfoil First, the stationary and oscillating behavior of an airfoil with NACA 0018 profile will be compared to a linear model. This model applies the thin airfoil theory which considers the airfoil as a two dimensional flat plate with zero thickness. The approach described by F.C. Fung [2] and M.A. Biot [5] is followed. This theory uses the acceleration potential theory. Other models, such as Von Karman and W.R. Sears [4], apply the velocity potential to determine the pressure difference of an oscillating airfoil. It is chosen to follow Fung. The experiments will be performed in a wind tunnel. A new method is used to simulate the non-steady motions of the airfoil. Instead of moving the wing with respect to the main flow, the flow will be oscillating while the wing is fixed. The flow will be disturbed by acoustic pressure waves, which are generated by two loudspeakers. On the test airfoil two transducers are mounted to determine the pressure difference over the airfoil. To obtain the stationary pressure difference the time average value is taken, for the oscillating difference a lock-in data analysis method is used. Attention is paid to theoretical difficulties, such as the finite thickness of the experimental wing, and to experimental difficulties, such as the acoustic velocity. As the acoustic velocity appeared a major parameter in the experiments, this parameter is considered in detail. Finally, in the NACA 0018 profile a cavity is made and the experimental results of this airfoil are compared to the airfoil without cavity, again both for the stationary and oscillating behavior. The influences of the amplitude of the acoustic perturbations and the angle-of-attack are considered as well. More, a coupling between flow instabilities and acoustic resonances is studied. This phenomenon is called airfoil self-noise. It is investigated if the cavity influences this coupling.


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1.3 Outline

The thesis is subdivided into the following chapters. Firstly, the theory of the thin airfoil is presented in chapter two. The problem is specified and the dynamical behavior of the stationary and oscillating airfoil is given separately. The experimental method, including the experimental setup and data analysis method, is considered in the third chapter. The fourth chapter considers the acoustic of the wind tunnel. A prediction of the acoustic field is made and subsequently verified by experiments. Using the results, the acoustic velocity is finally estimated. The fifth chapter verifies the measurement method by comparing the dynamical behavior of the NACA 0018 airfoil and the thin airfoil model. The stationary and oscillating behaviors are discussed successive. Several parameters, like Strouhal number, angle-of-attack, and amplitude of acoustic perturbations will be varied. Next, the sixth chapter considers the impact of the cavity on the dynamical behavior. The stationary and oscillating dynamical behavior of the airfoil with cavity is compared to the NACA 0018. The influences of the amplitude of the acoustic perturbations and the angle-of-attack are discussed. Finally, the conclusions and recommendations are presented in chapter seven. Results of the airfoil self-noise can be found in Appendix A.

1.4 Airfoil geometry

Along the thesis geometrical aspects of the airfoil are used, which are supposed standard knowledge. A short definition of some aspects will be discussed now. The geometry of an airfoil is given in figure 3. An airfoil has a round leading edge and sharp trailing edge. Between these two points a straight line can be drawn which is called the chord line. The length of this line is the chord, c , of an airfoil. An often used characteristic length is the half chord length, which will be indicated by the symbol b , i.e. bc 2= The angle-of-attack, α , is the angle between the chord line and direction of main flow, U . Finally, the camber line is the median line between the upper and lower surface of an airfoil and is useful to indicate the asymmetry of an airfoil. The camber is the maximum height between the camber line and chord line.

Figure 3 Geometry of an airfoil [3]


4


5

Chapter 2

Thin airfoil model An analytic model is presented which describes the aerodynamics of an airfoil. This model will later be used to verify the experimental method. Two situations are considered. In the fist situation, the airfoil performs a horizontal translation with uniform velocity (stationary motion). In the second case the airfoil moves non-stationary by means of an oscillating motion added on the steady translating motion. This situation imitates the small deformations at the surface of an airfoil in flight, which are caused by continuous forces. In linear theory these motions can be approximated as a superposition of harmonic oscillations. The model studies the influence of the stationary and oscillating motions on the local pressure differences along the airfoil. Firstly, the problem is specified and the boundary conditions are given. Next, mathematical techniques are explained which are used to solve the problem. Thereafter, the analytic models of the stationary and oscillating airfoil are solved. The approach is described by M.A. Biot [5] and Fung [2].

2.1 Specification problem

The problem of the stationary and oscillating airfoil will be specified. The flow around an airfoil is assumed to be two dimensional and incompressible. This

approximation is valid for low Mach numbers, 1)( 22 <<= cUM where U represents the

velocity of the flow and c the speed of sound. The main motion of the airfoil is a translation with constant velocity in x-direction, U . In the oscillating situation a second motion is introduced. This is a harmonic oscillation of small amplitude, i.e. the velocity associated to this oscillating motion ( ',' vu ) is assumed to be small

compared to the main velocity U . Therefore, the dynamics of the airfoil can be linearized and it is justified to use the principles of superposition to add the solution for an unsteady, harmonic oscillation on the steady-state solution for the main motion. To obtain an analytical solution a thin airfoil theory is applied which considers the airfoil as a 2D profile of zero thickness. In the stationary situation an angle-of-attack, α , is given to the airfoil. In the dynamical situation the angle-of-attack is zero. The chord length of the airfoil, b2 , is equal to two times unity and the midpoint is located at the origin of an xy-coordinate system, so the leading edge is positioned at x = -1 and the trailing edge at x = 1 for a zero angle-of-attack. The airfoil moves with constant velocity, U , in the negative x-direction. This is consistent to a fixed airfoil and a moving fluid in the positive x-direction with equal velocity, U . Figure 4 presents a schematic view of the stationary and oscillating situation.


6

(a) (b) Figure 4 Thin airfoil is performing (a) main motion (stationary) with fixed angle-of-attack, α , and (b) main and oscillating motion in y-direction (oscillating). Here the angle-of-attack is zero, α = 0. For both

situations the half chord length is equal to unity, b = 1, and the position of the airfoil is described by =y

),( txY .

Accordingly, the position of the stationary airfoil, y , is given by

xtxYy α−== ),( (2.1)

The oscillating airfoil performs a pure vertical translation. The position of the airfoil then is given by

tie ˆ),(' ωytxYy == (2.2)

Here y is a constant amplitude, ω the frequency of the oscillation in [rad/sec], t the time,

and 1 2 −=i . A complex notation is used1. Note that this specific motion is independent of the x-coordinate when the angle-of-attack is zero, 0=α .

2.1.1 Assumptions

To determine the dynamical behavior of an airfoil the following assumptions are made - The airfoil can be approximated as a horizontal plate

- Fluid is non-viscous ( 1Re >>= νUL )

- Flow is incompressible ( 12 <<M ) - Flow is isentropic - Fluctuations due to the oscillations, 'u and 'v are small compared to the main flow U

(| 'u |, | 'v | << U )

2.1.2 Boundary conditions

Boundary conditions are required to obtain a well posed problem. The thin airfoil needs two boundary conditions, moreover two additional conditions are required to determine the oscillating motion of the airfoil. The boundary conditions are listed below

1 Actually, [ ]tieytxY ˆRe),( ω= . For clarity it is not explicitly stated that only the real part of tiey ˆ ω is

considered.


7

I. The Kutta-Joukowski condition states that the velocity at the trailing edge remains finite. This condition implies that the flow leaves the trailing edge smoothly and parallel to the thin airfoil, i.e. the flow separation is tangent (see figure 5). This flow separation is induced by viscous forces. Hence, the Kutta-Joukowski condition incorporates an effect of viscosity within the framework of a frictionless theory [3], [8].

Figure 5 Flow pattern over a thin airfoil applying the Kutta-Joukowski condition.

IIa. On the surface of the airfoil the component of the velocity of the fluid normal to the airfoil has to be equal to the velocity of the airfoil, i.e.

nvnv fluidairfoil

vvvv⋅=⋅ (2.3)

For the oscillating motion of the airfoil some additional boundary conditions are required

IIb. When the boundary equation for the fluid velocity normal to the airfoil is linearized,

∂∂

<<∂∂

x

YU

x

Yu' , an expression of the vertical oscillating velocity of the fluid at the

airfoil, tievv

ω ˆ'= , will result

tieyi

x

YU

t

Y

Dt

DYv

ωω ˆ ' =∂∂

+∂∂

== (2.4)

The last step in (2.4) is valid for the particular situation where the motion is a rectilinear displacement and independent of the coordinate x . In the equation, DtD / represents the material derivative.

III. By differentiating the results with respect to the time, the boundary condition for the

acceleration, ti

yy eaaω

ˆ' = , of the flow in vertical direction is obtained.

tiy ey

Dt

YD

Dt

Dva

ωω ˆ'

' 2

2

2

−=== (2.5)

IV. At infinite distance, the oscillations in the flow field are damped. Hence, 'u and 'v → 0

when | x | or | y | → ∞ .

2.2 Mathematical techniques

To solve the problem of the stationary and oscillating thin airfoil, two mathematical techniques are used; acceleration potential and conformal mapping. The basic principles of these techniques are given in successive order. More detailed information can be found in appendix B.


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2.2.1 Acceleration potential

The kinematics of the flow field can be described by introducing an acceleration potential, φ , from which in each point the acceleration can be calculated by taking the divergence. Accordingly, φ∇=a

r (2.6)

It can be proven that this function satisfies the Laplace equation (Appendix B),

02 =∇ φ . (2.7)

Hence, when a two dimensional flow is considered, 0// 22222 =∂∂+∂∂=∇ yx φφφ , and the

acceleration potential exists and is differentiable in the whole domain, implies that a second function can be found which is orthogonal to this acceleration potential. This function is called the acceleration streamfunction, ψ. Together with the acceleration potential this function forms the complex acceleration potential, )(zw , which is an analytical function of the complex variable

jyxz += ,

ψφ jzw +=)( in which jyxz += and 12 −=j 2) (2.8)

Using the Euler equation for a incompressible and isentropic flow, it can be shown that the acceleration potential, φ , is directly related to the pressure (Appendix B). So, once the

acceleration potential is known the aerodynamic forces on an airfoil can be calculated.

0ρ

φp

−= (2.9)

Here, 0ρ is the density of the fluid at the reference state.

2.2.2 Conformal mapping

A conformal mapping method is used to transform the problem, together with the boundary conditions, into a problem for which solutions are already known. This method is based on complex function theory and is a mathematical tool to simplify problems such as the Laplace equation. A conformal mapping, )(zF , transforms a point in the complex z-plane, jyxz += , to a point

in the complex ζ-plane, ηξζ j+= . A property of such method is that two line elements in one

point are multiplied by the same factor when they are transformed, but the angle between the two line segments remains the same. Hence, the geometrical properties are preserved. (Appendix B).

2) Both i and j indicates an imaginary number, i.e. i 2 = -1 and j 2 = -1. However, both numbers are used for a different application. While i indicates a phase shift in time for a harmonic oscillation, j is used to separate the potential and its conjugate in the complex acceleration potential.


9

A common conformal mapping in aerodynamic studies is the Joukowski transformation in

which a straight line segment [ ]1,1−∈x in the z-plane, jyxz += , is mapped into a circle

with radius one in the ζ-plane, ηξζ j+= . The function is

( )

+== −

ζζζ

1

2

1z

1F (2.10)

This conformal transformation is used in the further calculations. A point located on the unit

circle ( θζ je= ) corresponds to the point θcos=x on the line segment where )/arctan( ξηθ = .

In each point on the circle the angle between the two adjacent line elements is preserved except at two points; the leading edge and trailing edge. These two points contain a singularity. The

space outside the circle in the ζ-plane is mapped into the entire z-plane. (See figure 6)

Figure 6 Joukowski Transformation

2.2.3 Boundary conditions

The acceleration potential has to satisfy boundary conditions. When the acceleration potential is

mapped into the conformal ζ-plane, the boundary conditions have to be transformed to the

conformal ζ-plane as well. This is presented in Appendix (C). Now the problem is specified, it can be solved by using the mathematical methods. First, the dynamics of the stationary airfoil will be determined, subsequently the oscillating airfoil.

2.3 Stationary airfoil

This section considers the dynamics of an airfoil translating in negative x-direction with uniform velocity, U . This is consistent to a fixed airfoil and a moving fluid in the positive x-direction with equal velocity, U . The airfoil is non-oscillating and thus stationary. The thin airfoil theory will be applied and the airfoil is assumed to be two dimensional and infinitely thin. The half chord length of the airfoil, b , is one and can therefore be omitted in this


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model. A fixed angle-of-attack, α , is given to the airfoil. The vertical position of the airfoil y ,

and the associated velocity, v , and acceleration, a , are for this situation given by (see figure 7) xxbtxYy ),( αα −=−== (2.11)

==

−=

∂

∂+

∂

∂==

02

2

Dt

YDa

Ux

YU

t

Y

Dt

DYv α

Figure 7 Thin airfoil with angle-of-attack, α . The stationary airfoil is fixed and the fluid flows with a

uniform velocity U in the positive x-direction. The half chord length is equal to one, b = 1.

The Joukowski transformation given in (2.10) is applied to map the airfoil to the ζ-plane. In addition, a proper complex acceleration potential, )(ζw , is needed.

2.3.1 Complex acceleration potential

For a uniform translating airfoil, a suitable complex acceleration potential, )(ζw , is proposed by

M.A. Biot [5]. The potential is in the ζ-plane,

( )b

jAw

+=

ζζ (2.12)

Here, A is a constant which has to be determined from the boundary conditions. The acceleration potential comprises a pressure dipole. A dipole can be obtained when a point source and point sink of equal strength are approaching each other in such a way that their strength multiplied with the separation distance is kept constant. The lines representing ψ = constant

(streamlines), are circles tangent to the axis of the dipole, as presented in figure 8.


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Figure 8 Acceleration streamlines generated by a pressure dipole

The acceleration potential proposed by M.A. Biot is presented in figure 9. The unit circle (bold

line) in the ζ-plane corresponds to the image of the airfoil. The dipole is placed tangent to the

unit circle in the ζ-plane and is located at point ξ = -1, which is the leading edge. One circle of

the arising acceleration streamlines of this potential coincides with the unit circle and contributes therefore nothing to the normal acceleration of the airfoil, thus a stationary motion is prescribed and no vertical oscillations are provided. Furthermore, a property of a dipole is that it contains a singularity at the origin of the dipole, that is to say, a point where the pressure tends to infinity and x∂∂ /ψ is undetermined is located at the leading edge.

The Kutta-Joukowski boundary condition states that the velocity is finite at the trailing edge. An equivalent form of this condition is that no pressure discontinuity exists at the trailing edge. Hence, the acceleration potential should be continuous at this point, ξ = 1. This condition is

satisfied.

Figure 9 Streamlines belonging to the complex acceleration potential, )1/()( += ζζ jAw . The potential

describes a stationary airfoil in a uniform flow along the ξ-axis. In the figure the streamlines are represented by thin lines, the thick circle corresponds to the airfoil.

The complex acceleration potential, )(ζw , consists of a real part, which is the acceleration

potential, φ , and an imaginary part, which is the acceleration streamfunction, ψ .


12

1

)(+

=+=ζ

ψφζjA

jw (2.13)

Where 11 /)sin( rA θφ = and 11 /)cos( rA θψ =

The polar coordinates 1θ , and 1r are explained in figure 10. The circle corresponding to the

airfoil is given by )cos(2 11 θ=r and 1θ ∈ [0, 2π].

Figure 10 Definitions of 1θ , and 1r .

2.3.2 Flow field

To determine the flow field, the constant A needs to be determined. The constant A can be determined by using the kinematic boundary conditions. Since the acceleration can be expresses as xDtDva ∂−∂== // ψ , the velocity of the flow is a function of

the streamfunction, ψ ,

xx

vU

t

v

Dt

Dv

∂

∂−=

∂

∂+

∂

∂=

ψ (2.14)

In this particular situation where the motion of the airfoil is stationary, the “ tv ∂∂ / ”-term vanishes. Integrating both sides to the x-coordinate results in

U

vψ

−= (2.15)

On the airfoil, )cos(2 11 θ=r , the streamfunction ψ is a constant.

( ) ( )

2)cos(2

coscos

1

1

1

1 AA

rA ===

θθθ

ψ (2.16)

Accordingly, from (2.11), (2.15) and (2.16) the constant A is expressed as


13

222

UAU

AUv αα =⇒−=−= (2.17)

Hence, the acceleration potential can be given as

( ) ( )

1

12

1

1 sin2

sin

rU

rA

θα

θφ == (2.18)

2.3.3 Stationary pressure difference

After the acceleration potential is known, the pressure distribution over the thin airfoil can be determined. From (2.9) the acceleration potential in an incompressible and isentropic flow is a function of the pressure, p , and density, ρ .

0ρ

φp

−= (2.19)

Consequently, the pressure distribution along the stationary airfoil, )cos(2 11 θ=r , can be given

as

[ ] [ ] )2/tan(2

0)cos(20)cos(2 1111θαρφρ θθ Up rr −=−= == (2.20)

Here, the coordinate 1θ is expressed in terms of θ , i.e. 2/1 θθ = (inscribed angles, see figure

10).

The local pressure difference, [ ] )cos(2 11 θ=∆ rp , over an airfoil is equal to the pressure differences

between the upper and lower side of the thin line segment. As the direction of the pressure is opposite to motion of the airfoil and since the airfoil is symmetric, the local pressure difference

is equal to [ ] [ ] )cos(2)cos(2 11112 θθ == −=∆ rr pp . So, in the ζ-plane

[ ] )2/tan(22

0)cos(2 11θαρθ Up r =∆ = (2.21)

It is seen that when the angle-of-attack is zero, 0=α , no pressure difference over the airfoil is experienced, in other words, an angle-of-attack induces a pressure difference. The curve of a dimensionless local pressure difference for an angle-of-attack of 5=α degrees is presented in figure 11. The x-axis represents the x-position along the airfoil divided by the half chord length b , the y-axis represents the pressure difference divided by the dynamic pressure,

22

1 UP ρ= . The largest pressure difference can be found close to the leading edge. At the

leading edge itself the pressure difference becomes infinitely as it contains a singularity. At the trailing edge the pressure difference is reduced to zero, which is a result of the Kutta-Joukowski condition.


14

-1 0 1 0

1

2

3

4

5

6

x / b

∆ p

airfo

il / (

1/2

ρ U

2)

Figure 11 Theoretical local pressure differences along a stationary airfoil, airfoilp∆ , for a fixed angle-of-

attack, α = 5 degrees.

2.4 Oscillating airfoil

In the oscillating situation the airfoil performs, next to the uniform displacement in x-direction, also an oscillating motion in y-direction. The induced aerodynamic forces on the airfoil due to these motions will be considered in this section. The vertical translating motion of the 2D oscillating airfoil is described by

titibeyeyty

ωω0

ˆ)(' == , where b is the half-chord length of the airfoil (which is one in this

model and is therefore omitted), 0y the ratio between the amplitude y of the vertical

displacement and half chord length b ( byy ˆ0 = ), and ω the angular frequency of the

oscillation. The latter one is often rewritten in terms of the dimensionless Strouhal number St , which is defined as the product of half chord length and angular frequency divided by the wind velocity;

UU

bSt

ωω== (2.22)

This ratio is sometimes also known as the reduced frequency [4]. For the non-stationary airfoil the angle-of-attack, α , is zero. Consequently no pressure difference is induced due to this angle. The position of the airfoil, and both the associated velocity and acceleration of the fluid at the surface of the airfoil, are given as

tieytxYy

ω0),(' == (2.23)

=−==

===

tiy

tiy

titi

eaeyStUDt

YDa

eveyStiUDt

DYv

ωω

ωω

ˆ'

ˆ '

022

2

2

0

Again the Joukowski transformation is used as mapping method. Subsequently, a proper acceleration potential is selected which satisfies the oscillating boundary conditions. Finally, after


15

determining some constants, the flow field is known and the pressure differences across the airfoil can be determined.

2.4.1 Complex acceleration potential

A suitable complex acceleration potential, tieww

ω ˆ= , to describe the acceleration of the flow around the oscillating airfoil is proposed by Fung [2],

ζζ

ζjBjA

w ++

=1

)(ˆ (2.24)

Here A and B are constants to be determined by the boundary conditions. Both terms of the acceleration potential are presented in figure 12, were the unit circle is the image airfoil. This complex acceleration potential consists of two pressure dipoles, one located tangent to the unit circle at point 1−=ξ , the other parallel to the first but located at the centre of the circle.

Figure 12 Streamlines belonging to the complex acceleration potential which consists of two pressure

dipoles, )1/( +ζjA , which corresponds to a stationary airfoil in a uniform flow along the ξ-axis, and

ζ/jB , which determines an oscillating airfoil. In the figure the streamlines are represented by thin lines,

the thick circle corresponds to the unit circle which is related to the airfoil.

The first term of the potential, )1/( +ζjA , is equal to the acceleration potential for a stationary

airfoil in a uniform flow (equation (2.13)). This potential provides a translating motion in ξ-direction but does not account for vertical oscillations. The second term of the acceleration potential, ζ/jB , is therefore added [5]. From the arising circular acceleration streamlines it is

seen that the streamlines cross the unit circle at different angles. Consequently, at each point a normal component of the acceleration vector on the unit circle exists and the oscillating motion is prescribed, with the exception of the leading edge (which contains a singularity) and the trailing edge. The first as well as the second term of the complex acceleration potential are continuous at the trailing edge, so the Kutta-Joukowski condition is satisfied.

According to (2.8) the complex acceleration potential, tieww

ω ˆ= , consists of a real part, which

is the acceleration potential, tie

ωφφ = , and imaginary part which is the associated

streamfunction, tie

ωψψ ˆ= .


16

ζζ

ψφζjBjA

jw ++

=+=1

ˆˆ)(ˆ (2.25)

In which

ˆˆˆ21 φφφ += 21

ˆˆˆ ψψψ +=

1

11

)sin(

1Reˆ

rA

jA θζ

φ =

+=

1

11

)cos(

1Imˆ

rA

jA θζ

ϕ =

+=

r

BjB )sin(

Reˆ2

θζ

φ =

=

rB

jB )cos(Imˆ

2

θζ

ϕ =

=

The meaning of the polar coordinates 1θ , 1r and θ , r is explained in figure 13.

Figure 13 Definitions of 1θ , 1r ,θ and r

2.4.2 Flow field

To determine the flow field, the constants A and B need to be calculated. Using the boundary conditions for the acceleration and velocity, the values of the constants A

and B can be obtained. The calculation is given in detail in Appendix D.

)( 2 02

StCyStiUA −= (2.26)

Where )(StC is the Theodorsen’s function (Appendix D). This function comprises the

zero and first order of a modified Bessel functions, 0K and 1K , and is given by

)()(

)()(

01

1

ikKikK

ikKkC

+= .

022yStUB = (2.27)

Consequently the acceleration potential, tie

ωφφ = , is known. In the ζ-plane this potential is

given by


17

( ) ( )

( ) ( )r

yStUr

StCyStiU

rB

rA

θθ

θθφφφ

sinsin )( 2

sinsinˆˆˆ

022

1

10

2

1

121

+−=

+=+= (2.28)

2.4.3 Oscillating pressure difference

As soon as the flow field is known the pressure difference between the top and bottom side of the airfoil can be determined. This is a measure for the dynamics of the airfoil. From (2.9), the acceleration potential is for an incompressible and isentropic flow a function of the pressure, p ,

and density, ρ .

0ρ

φp

−= (2.9)

Hence, the local oscillating pressure difference, 'p∆ , at the airfoil, 1=r and )cos(2 11 θrr = , is

equal to

[ ] [ ][ ]

( ))sin()2/tan()( 2

2

'2'

20

20

1 0

1 1

θθρ

φρω

ω

StStCiSteyU

e

pp

ti

r

ti

rr

+−=

=

−=∆

=

==

(2.29)

Consequently, the total oscillating force, tieFF

ω ˆ'=r

, can be obtained by an integration of the local pressure difference. Note that the pressure difference is specified in the ζ-plane, as a result the integral has to be adapted to this plane. In the z-plane the integral would be taken over the line segment between x = -1 and x = 1. In the ζ-plane, this corresponds to an integration with

respect to θθ ddx )sin(= between the boundaries θ = 0 and θ = π

[ ] [ ]

−=

+−=

+−=

∆=∆=

∫∫

∫∫ =

−

=

)(2

1

2 )( 2

)(sin )sin()2/tan()( 2

)sin(' 'ˆ

022

0

20

20

0

22

0

02

0

0

1r

1

1

1r

StCSt

iyStUπρ

StStCiStyU

dStdStCiStyU

dpdxpF

ππρ

θθθθθρ

θθ

ππ

π

(2.30)

Transforming this result back to the z-plane the value remains the same, as the total force is

independent of the polar coordinates. So, the total force, tieFF

ω ˆ'=r

, of an oscillating airfoil is given by

−= )(2

1022

0 StCSt

iyStUπρF

) (2.31)


18

According to (2.31), the force consists of two terms. The first term can physically be interpreted as a force caused by the inertia of the apparent mass of the flow around the airfoil. It is the

product of the acceleration, tieyStU

ω0

22 , and the apparent mass of the flow around a line

segment, which is πρ for the situation where the half chord length is equal to unity, 1=b . The

force acts in the mid-chord point of the airfoil. The second term can be split up into two forces, a stationary part and fluctuating part. Both forces can be explained by the vorticity theory, which put forward that during a flight a circulation around the airfoil arises. This circulation is called the bound vortex and results to a force. The stronger the circulation is, the stronger the force will become. The stationary part is due to the vorticity which is generated at the beginning of the flight. It is the force which the airfoil experiences if the main velocity and angle-of-attack are maintained. When not, additional vortices are created in the wake which also influence the total force, this is the fluctuating part. In general, the bound vorticity is the resultant of an airfoil in flight. Moreover, the force due to the bound vorticity acts on ¼ of the total chord length and induce therefore a moment about the mid-chord [4]. Since the angle-of-attack is zero, the arising forces given in (2.31) are purely the result of the oscillating motions. For the oscillating airfoil the curve of the local pressure difference across the airfoil is presented in figure 14. The pressure differences given in (2.29) are plotted for different values of the Strouhal number, UbSt /ω= . The x-axis represents the x-position of the airfoil divided by the half chord length b , the y-axis represents the pressure difference divided by the dynamic

pressure, 22

1 UP ρ= . From the figure it is seen that the pressure difference increases in the

middle of the airfoil for increasing Strouhal number. The pressure difference becomes infinite at the leading edge, which contains a singularity point, and is zero at the trailing edge. The latter point is the consequence of the applied Kutta-Joukowski condition, which provides tangential flow separation at the trailing edge.

-1 0 1x/b

∆ p

' airfo

il / (

1/2

ρ U

2 )

St = 4

St = 3

St = 2

St = 1

St = 0.5

Figure 14 Theoretical local pressure differences along an oscillating airfoil, airfoilp'∆ , for several values of

Strouhal number, St . The airfoil performs pure translational oscillations. Angle-of-attack is zero, α = 0


19

Chapter 3

Experimental method In this section the principle of the experimental method is described which measures the dynamical behavior of the airfoil. A local pressure difference between the top- and bottom sides of the airfoil provides information on the dynamic response. The steady pressure differences are estimated from the time averaged pressure differences and the dynamical by the oscillating ones. The experiments are performed in a wind tunnel, which is described in the first part of this chapter. To investigate the oscillating motions a new technique is introduced. Instead of mounting the wing on a complex suspension and moving the wing with respect to the main flow, the flow will now be oscillating while the wing is fixed. The data analysis method is the lock-in method, which is explained in the second section.

3.1 Experimental setup

This section discusses the experimental setup which measures the stationary and oscillating pressure difference over the airfoil. The experiments are performed in a low speed wind tunnel. The wind velocity in the test section, U , ranges between 0 m/sec and 60 m/sec. The wind tunnel is a closed circuit. Firstly, the flow passes the large settling chamber where the flow is straightened. After the settling chamber the flow passes through a cone which contracts the flow and accelerates it. Subsequently a uniform flow will go through the test section. Behind this section a gap is found which equalizes the pressure between the test section and the surroundings. Finally, a diffuser is used to decelerate the flow and to recover the kinetic energy as potential energy (pressure). The flow is driven by a ventilator, which is located beneath the floor (see figure 15).


20

Figure 15 Geometry of the wind tunnel. The flow passes in subsequent order the Settling chamber, test section, gap, and diffuser. The flow is driven by a ventilator which is located in the return channel beneath the floor.

The velocity in the test section, U , is determined by measuring the pressure difference, p∆ ,

between the settling chamber and the inlet of the test section. The pressure difference is measured by means of a MU DIGITAL differential pressure transmitter. The velocity is determined by applying the equation of Bernouilli, in which the velocity in the settling chamber is neglected

22

1 Up ρ=∆ (3.1)

The velocity is acquired with an accuracy of 1.6%.

3.1.1 Test section

The test section is made of multiplex and has a size of 500mm x 500mm x 1000 mm (see figure 17 and 18). The thickness of the walls is 24 mm. The airfoil is clamped in the middle of the test section by a pin-hole-connection to the floor of the wind tunnel, and by a rod with a flange at the top side. Consequently, rotations are allowed with an accuracy of 0.5 degree. To investigate the stationary motion the pressure difference over the airfoil is determined as function of the angle-of-attack. The dynamical motions are induced by a transversal oscillating acoustic flow. To generate these perturbations two Monacor speakers (200W, max 8 Ω) are placed opposite to each other, in the middle of the test section. To avoid a pressure difference across the speaker (the pressure difference between the test section and ambient), a box is placed over the speaker on the outside of the wind tunnel. A detail sketch of the speakers mounted on the test section is given in figure 16.


21

Figure 16 Mounting of a speaker on one side wall of the wind tunnel. Around the speaker a box is placed to avoid the pressure difference across the speaker.

The speakers are tuned to the first transversal resonance frequency of the wind tunnel to create a transversal standing wave. The speakers are connected in phase, so that the acoustic waves reinforce each other. The driving electrical signal is generated by a function generator (type: Yokogawa FG120). Subsequently the signal is amplified by a power amplifier (type: WPA 301A) before it is sent towards both speakers. Characteristic of this experiment is that the frequency is limited to the transversal resonance frequencies of the wind tunnel. Accordingly, only relative high Strouhal numbers can be achieved since the first transversal resonance frequency is rather high (340 Hz, this will be discussed in Chapter 4). Hence, the oscillating pressure differences on the airfoil are dominated by inertia forces of the apparent mass. The wall of the wind tunnel has several openings in which piezo pressure transducers can be placed. This is done to investigate the acoustic field and to determine the acoustic velocity at the airfoil. The used pressure transducers are two PCB (type 116A) and three Kistler (type 7031). Firstly, two holes are made at the top, between the trailing edge and speaker, and two PCBs are plugged in (position I and II, figure 18). Later, it appeared that more information was required and five additional openings have been made; three in upstream direction on the top wall (position A, B, and C), and two in the middle on the side walls (position D and E). In these additional holes the three Kistler piezo transducers can be placed. The remaining holes are closed by means of a dummy plug. The sensitivity of the transducers [Pa/pC] is provided by the calibration carried out by the manufacturer (accuracy 1%). The output signals of the PCBs are amplified by means of Kistler type 5011 amplifiers, the output of the Kistler pressure transducers by Kistler type 5007 amplifiers. The sensitivity is set at 1 pC/V for all amplifiers. An overview of the test section of the wind tunnel is given in figure 17 and 18. A photo can be found in Appendix F.


22

Figure 17 Experimental setup; 1) wind tunnel section, 2) speakers, 3) location PCB pressure transducers 4) location Kistler pressure transducers 5) wing without cavity, and 6) Kulites pressure transducers. The arrow (7) represents the direction of the main flow, U . The boxes over the speakers are left out.

Figure 18 Top view experimental setup. The positions of the pressure transducers are indicated by “•”. The PCB’s are placed at the top of the wind tunnel (I and II), the Kistler transducers at the top of the wind tunnel (A, B, and C) or at the side walls (D and E). In the middle the location of the airfoil is given which is placed between the two speakers. The main wind velocity is directed from left to right. All dimensions are given in mm.

3.1.2 Experimental wing

Next, details of the experimental wing are considered. A first series of experiments are performed on an aluminum extruded wing without cavity. The experimental wing has a NACA 0018 profile (Appendix E). The profile is symmetric, which is specified by the double 0 in the code. Furthermore, the profile is characterized by its relative


23

thick profile (18% of the chord length). A thick profile is typical for HALE wings. The chord is c = 165 mm, the thickness is d = 29,7 mm, and the span width is w = 495 mm. On the experimental wing two pressure transducers are mounted to measure the local pressure difference of the airfoil 3. One is placed on the upper side and one on the lower side of the airfoil (see figure 19). The transducers are Kulite semiconductors (type: XCS-093-140mBarD). These pressure transducers record both the static and dynamic pressure, whereas the PCBs and Kistlers can only measure the dynamic pressure. Also the Kulites have a diameter of only 2 mm while the piezo transducers have a diameter of about 1 cm. On the other hand, the PCBs and Kistlers have a better sensitivity than the Kulites. The calibration and linearity of the Kulites provided by the manufacturer are checked statically by comparison of the transducer output to that of a precision water micro-manometer (type Betz, accuracy 1Pa).

Figure 19 NACA 0018 airfoil. Position of Kulites k81 and k83 on the airfoil are marked by “ ”. Dimensions are given in [mm].

Secondly, experiments are performed on an airfoil with cavity. In the NACA 0018 airfoil a cavity is made along the span width of the wing. The cavity has a round cross-section with a diameter of 32 mm. The cavity is milled at an angle of 20 degrees. The geometry of the cavity is rather arbitrary but is inspired by the geometries proposed by the European project VortexCell 2050. Inside the cavity a third pressure transducer is placed (see figure 20)3.

Figure 20 NACA 0018 profile with cavity. Kulites k9, k12 and k84 are marked by “ ”. Dimensions are given in [mm].

Photos of the experimental wings can be found in Appendix F.

3 Additional Kulite pressure transducers are placed on both airfoils. The NACA 0018 airfoil contains an additional transducer at the leading edge. The airfoil with cavity contains one extra at the leading edge and one in the cavity. However, the results of these transducers are not used in this thesis and are therefore left out.


24

3.1.3 Data acquisition

Data acquisition is managed by a SCXI-module of National Instrument (no. 1000), consisting with an 8-channel input (no. 1180). The program Scope Corder (made in Labview) is used to record the data. Each channel is recorded independently and a cut-off frequency and sensitivity can be set to each channel. The sample rate is 10 000 samples per second and the measuring time is around 15 seconds. The recorded data is subsequently analyzed using a lock-in method.

3.2 Data analysis: the Lock-in method

The stationary pressure difference across the airfoil is obtained by a time-averaged signal. In order to extract the oscillating pressure difference from the signal a lock-in data analysis method is needed. The lock-in method is a powerful method to extract a weak signal at a specific frequency out of a noisy background. The method is based upon the orthogonality of sinusoidal functions. The basic principle of the lock-in method, the practical application, as well as the approximated error will be discussed. More detailed information about the mathematical principles and Hilbert transform, which is used for the lock-in method, can be found in the appendix G.

3.2.1 Basic principle

The basic principle of the Lock-in method relies on the property that an integral over a sinusoidal function multiplied to another sinusoidal function with different frequency will vanish, while an integration of the product of two sinusoidal functions with equal frequency results to a value. Optimal results are obtained when the integral is taken over an integer number of time periods. That is to say, the response signal of the experimental system is first multiplied by a reference signal, which corresponds in this case to the driving signal to the amplifier which in turn drives the speakers. Integrating the multiplied signals will result to a value determined by the Fourier components of the signal at the reference frequency. Other signals, such as noise, consist of different frequencies and will therefore strongly be attenuated. A block diagram is given in figure 21. The output signal of the lock-in method is represented by )(tg . It can be said that this

signal is “locked” to a coherent reference signal, )(tr [12]. A precondition is that only the

information at one specific frequency is of interest.

Figure 21 Block diagram of Lock-in method implemented in an experiment

A big advantage of the lock-in method is that a desired signal can be extracted from measurements containing a high level of noise. It provides measurements at a very narrow bandwidth, while other methods, like a narrow band-pass filter, are limited in practical use. For instance, a narrow-pass filter has problems when frequency shifting or zero drifting occurs. In a lock-in method the signal follows the reference signal, )(tr , and overcomes this problem [15].


25

3.2.2 Practical application

This section considers the practical application of the lock-in method in the experiments. In the experimental setup the reference signal, )(tr , is the driving signal to the amplifier of the

speakers. This signal is produced by the signal generator and is a sinusoidal function with constant frequency and amplitude. The response signals, )(tf , are the signals from the pressure

transducers on the airfoil as well as from the transducers in the wall of the wind tunnel. In Matlab an algorithm has been written which uses the lock-in method to extract the amplitude of the mode at the reference frequency and the phase difference between this mode and the reference signal out of the response signals. Firstly, the time average is removed from all the signals. As a result only the amplitudes of the oscillating modes will be maintained. Subsequently, an integer number of oscillating periods is selected. For this the first data points of one oscillation of the reference signal, )(tr , are taken. The point is searched which starts (close

to) the zero line and intersects this line at a positive slope. This is the initial point of the integration period. A tolerance domain, δ , is set in which this initial point should be located. This tolerance is defined as the maximum increment between two adjacent data points. The positive slope is prescribed by a positive difference between two adjacent data points. The process is explained in figure 22. The same is done to obtain the end point of the integration. Here, the data points for the last oscillation are taken. Accordingly, all the data points of the signals in this range between the initial and end point are selected.

1 5 10 15 20 25 30-3

-2

-1

0

1

2

3

no. data point [#]

Am

plit

ude r

efe

rence s

ignal [V

]

δ

Figure 22 Process to select initial point for the integration from the reference signal. The data points located in the tolerance domain, δ , are selected. The initial point is the one which intersect the zero line

with a positive slope. This point is represented by “”.

Next, the reference signal, )(tr , is multiplied by a scale factor to make the amplitude equal to

unity. This scale factor is one divided by the absolute time averaged amplitude of the reference signal. As a result, when the product is taken of the response and reference signal, the outcome is equal to the amplitude of the response signal. The frequency of the reference signal is determined from the time rate of change of the instantaneous phase angle. This value is averaged over the time.


26

The output signal of the lock-in method, )(tg , can be decomposed in two sinusoidal functions

a and b , in which the second signal is ½ π phase shifted to the first one (first order Fourier series, see appendix G). In the lock-in method, the angular frequency of the functions is equal to

the reference frequency, refωω = where f 2πω = . The first signal, a , is determined from the

integration of the product of the response signal, )(tf , and reference signal, )(tr , whereas the

second function, b , is determined from the integration of the product of the response signal,

)(tf , and the ½ π shifted reference signal, )(* tr . To obtain this ½ π shift a Hilbert

transformation is used (appendix G). Hence, functions a and b are equal to )(1 traa = and

)(*1 trbb = , where the constants 1a and 1b are the result of the integration and contain

information about the amplitude and phase difference of the response signal, )(tf and reference

signal, )(tr . When both signals, a and b , are determined the output signal, )(tg , is known.

[ ][ ])(*)(Re

)()(Re)(

11 trbtra

tbtatg

+=

+= (3.2)

The reference signal, )(tr , does not have to have a perfectly constant amplitude and frequency,

which is an advantage of the method. In this thesis the reference signal can be approximately as sinusoidal function of the time so that

[ ]

+=

−=

+ tibaieeba

tbtatg

ωπ

ωω

))/(arctan(21

21

11

11Re

)cos()sin(Re)(

(3.3)

Equation (3.3) will be used to determine the amplitude and phase of the signals with respect to the reference signal.

3.2.3 Error

In this lock-in analysis method two errors occur, which are considered next. Firstly, an error appears when the integration period is not exactly an integer number of oscillation periods. Due to digitalizing the measuring data this integration period may deviate. Secondly, an error is created due to noise that happens to be at the specified frequency. Increasing the measurement time is a method to reduce these problems. For the first case the

error will be reduced by a factor of 1/N, for the second situation by 1/√N if the phase of the noise is random. Here, N represents the number of periods. Figures 23 and 24 represents the output signal, )(tg , of the lock-in method for the pressure

transducer at the top of the airfoil (Kulite k81). Figure 23 gives the amplitude in [mV] as a function of the measuring time in [sec]. In one situation the integration is taken over an integer number of oscillating periods (solid line), in the second situation the integration is taken over an arbitrarily chosen time interval (dotted line). Figure 24 shows a detail of the data signal for the output signal, )(tg , and the original response signal, )(tf .

From the figure it is seen that just after 10 to 20 time periods the error due to a non-integer number of time period is faded away. Error due to noise needs more time to diminish, roughly 400 oscillating periods, or about 1 second when the frequency is 340 Hz. This is quite fast. In


27

this case, the relative error for the Kulite semiconductor is 0.49%. In the experiments a measuring time of 15 seconds is used, the accuracy is 0.23%. The same is done for the transducer in the wind tunnel wall (piezo transducer PCB). The error for this signal is 0.15%. The results can be found in Appendix H.

10-3

10-2

10-1

100

101

102

103

0.2

0.25

0.3

0.35

0.4

0.45

measuring time [sec]

am

plit

ude [m

V]

Kulite k81

with integer periods

arbitrarily time interval

0.405

Figure 23 Investigation inexactness lock-in method of a pressure transducer on the top of the airfoil (kulite k81). The output of the lock-in method is given for two situations, one is the lock-in method taken over an integer number of periods (solid line), and the other the method taken over a random time interval (dotted line). The x-axis gives the measuring time in [sec] on a log-scale, the y-axis gives the amplitude in [mV]. The main velocity in the wind tunnel is 41 m/sec.

400 450 500 550 600 650 700-0.4

-0.2

0

0.2

0.4kulite k81

no. data point [#]

am

plit

ude [m

V]

response signal, f(t)

output signal, g(t)

Figure 24 Detail of the data signal of pressure transducer on the top side of the wing. The response signal, )(tf , and the output signal of the lock-in method, )(tg are given. The

measurement is taken at a mean velocity of 50 [m/sec].


28


29

Chapter 4

Acoustics of the wind tunnel The oscillating flow around the airfoil is induced by acoustic waves, which are driven by two speakers in the side walls of the wind tunnel. The acoustic velocity is an important experimental parameter, therefore it is important to determine the imposed acoustic perturbations quantitatively. This chapter studies the acoustics of the wind tunnel. Firstly, a theoretical analysis of the acoustics is given. In the second part the acoustical field is experimentally determined and the results are compared to the theory. Finally, the transversal acoustic velocity is estimated.

4.1 Theoretical background

This section discusses the theory of the acoustics in the wind tunnel. Firstly, the basis equations of acoustic waves are given. Subsequently, the equations are applied to a wind tunnel section. Hereafter, the results are revised for the case of a uniform velocity in the wind tunnel. Finally, the theoretical acoustic velocity is calculated.

4.1.1 Basic equations

Acoustics describes the evolution of perturbations in a stationary, reference fluid. This reference

fluid is given by a uniform density, 0ρ , and pressure, 0p . For now it is assumed that the

reference fluid is at rest, i.e. 00 =Vr

. In a later calculation a velocity will be introduced. The

perturbations in the fluid are described by ' ρ , ' p , and 'vr

, accordingly the density, pressure and

velocity of the fluid are considered as '0 ρρρ += , '0 ppp += and 'vvrr

= respectively. In

acoustics the density perturbations, ' ρ , are assumed to be small compared to the density of the

reference state of the fluid, i.e. 0' ρρ << .

When the time derivative of the mass conservation law is taken, and divergence of the momentum equation is subtracted from it, a hyperbolic partial differential equation is obtained.

When subsequently a thermodynamic relationship is used, ρρρ dcdpdp s20)/( =∂∂= , and the

equation is linearized, an equation arises which describes the propagation of a pressure disturbance. This equation is known as the homogenous wave equation [18].

0''1 2

2

2

2

0

=∇−∂

∂p

t

p

c (4.1)


30

Where 0c is defined as the speed of sound in the reference fluid: s

pc

∂∂

=ρ

2

0 . The subscript

“s” denotes that the derivative is taken at constant entropy.

A plane wave is considered. Characteristics of this wave is that its parameters, ' p , ' ρ and 'vr

are uniform in a flat surface normal to the direction of propagation. For plane waves propagating in the x-direction, the wave equation in (4.1) reduces to

0''1

2

2

2

2

2

0

=∂

∂−

∂

∂

x

p

t

p

c (4.2)

A general solution of this equation has been given by d’ Alembert [18].

)()('00 c

xtg

c

xtfp ++−= (4.3)

Here f represents a pressure disturbance which translates in the positive x-direction. Likewise,

g is the pressure disturbance moving in the negative x-direction. The functions f and g are

determined by the initial and boundary conditions. The shapes of pressure disturbances f and

g do not change in time, but only translate through space with speed 0c .

Assume a harmonic wave. This kind of perturbation is written in complex notation

as [ ]tie ˆRe' ωpp = , where p is the complex amplitude, ω the angular frequency, t the time, and

i the imaginary unit, 12 −=i . Then the solution of d’ Alembert can be written as

[ ]

+==

+−

−+

)()(00Re ˆRe'

c

xti

x

c

xti

xti

epepeppωω

ω (4.4)

or

[ ])()( 00Re'xkti

x

xkti

x epepp+−−+ += ωω (4.5)

Where 0k is the wavenumber, λπω /2( / 00 == ck , in which λ is the wavelength). Moreover, +xp is the amplitude of the wave traveling in the positive x-direction and −

xp is the amplitude of

the wave traveling in negative direction (see figure 25). The acoustic velocity perturbation in x-direction, 'u where ]',','[' wvuv =

r, is related to the

acoustic pressure disturbance, 'p , by the linearized momentum equation in x-direction,

0/')/1(/' 0 =∂∂+∂∂ xptu ρ . Using the result of the pressure disturbance in (4.5), the relation of

the acoustic velocity becomes

( ) [ ])()()()(

00

0000 Re1

Re'xktixktixkti

x

xkti

x eueuepepc

u+−−++−−+ +=

−= ωωωω

ρ (4.6)


31

where 00c

pu x

ρ

++ = , and

00c

pu x

ρ

−− −

=

Figure 25 Top view of the wind tunnel. Plane waves are propagating in the x-direction of the wind tunnel.

4.1.2 Pipe modes in wind tunnel

Next, the acoustical basic equations are applied to a wind tunnel section. In a straight pipe, plane waves always propagate along the pipe axis. In the situation that the wavelength is larger than twice the width of the wind tunnel, that is to say the frequency is below a cut-off frequency, these waves are the only propagating waves. At higher frequencies more complex waves can travel along the pipe. By using the basic equations the structure of these higher order modes can be identified and information about the cut-off frequency can be obtained. The geometry of the wind tunnel is presented in figure 26. The wind tunnel has a square cross-section with width L2 and is assumed infinitely long, i.e. no boundary conditions are prescribed to the x-direction. The x-axis is parallel to the midline of the duct, and the yz-plane corresponds to a plane normal to this axis, so Ly −= , Ly = and Lz −= , Lz = corresponds to the walls of

the wind tunnel.

Figure 26 Geometry of the wind tunnel section

An acoustic wave in the wind tunnel has the form of a harmonic mode (Appendix I). The pressure of the mn -th mode is given by


32

( ) ( )( )[ ]

[ ] Re

zsin y sinRe'

)(

)(

2

1)-(n

2

1)-(m

xkti

mnx

xkti

mnx

tixik

mnx

xik

mnxnzmymn

mnxmnx

mnxmnx

epep

eeAeAkkp

+−−+

−−+

+=

+++=

ωω

ωππ

(4.7) in which for positive integer m and n

0k ; wavenumber; 00 / ck ω=

L

mk my

2

π= ; wavenumber of the m -th mode in

y-direction

L

nk nz

2

π= ; wavenumber of the n -th mode in

z-direction

( )22

2

20

2nm

Lkk mnx +

−=π

; wavenumber of the mn -th mode in x-

direction

±mnxA ; Amplitude of the mn -th mode in the positive

or negative x-direction, indicated by the ± sign. The value has to be determined by the boundary conditions.

The amplitude of the mn -th mode traveling in positive x-direction is represented by +mnxp , and

for the negative x-direction by −mnxp . The overall solution for the wind tunnel is a superposition

of the different modes,

∑∑∞

=

∞

=

=0 0

''n m

mnpp (4.8)

Two situations can occur:

i) The wavenumber mnxk is real when the frequency 21220 4 /

)nL)(m/(cf +> , in which

f 2πω = . In this situation the mode propagates along the wind tunnel. Nevertheless,

the phase velocity, c , is not equal to the speed of sound, 0c , but is equal to

mnxkc /ω= .

ii) In the second situation the wavenumber mnxk is imaginary. Here the frequency of the

perturbation is smaller than 2/1220 ))(4/( nmLc + and amplitudes of the modes are

attenuated or amplified. This decay or amplification is exponential. These modes are called evanescent.

Accordingly, the cut-off frequency of the wind tunnel is equal to


33

2/1220 )(4

nmL

cfc += . (4.9)

For the first non-planar wave ( 0 ,1 == nm ) the cut-off frequency is equal to )4/(0 Lcf c = . If

the wavelength of a wave is larger than two times the width of the tunnel a first transversal mode can propagate, while for lower frequencies the mode is evanescent [18]. The resonance frequency for this first transversal mode arises when a half wavelength fits to the

width of the wind tunnel, i.e. L4=λ . This corresponds to λππ /2)2/(1 == Lk y , which is

equal to 0k . Then, the first transversal resonance of the pipe is

L

cf res

4

0= (4.10)

and the wavenumber in x-direction is

010 =xk (4.11)

The cut-off frequency is equal to the transversal resonance frequency. Note that 010 =xk

implies that the phase speed in the x-direction is infinite, i.e. there is no propagation of energy in the x-direction associated with this wave (the group velocity vanishes). To obtain high amplitude perturbations, the speakers in the wind tunnel should be tuned to the first transversal resonance frequency of the tunnel ( 0 ,1 == nm ). The width of the experimental

wind tunnel is L2 = 0.5 m and the corresponding resonance frequency of the first transversal

mode is resf = 344 Hz. Here, the speed of sound in air, 0c = 344 m/sec, is taken at the

laboratory temperature, T = 293 K.

4.1.3 Uniform velocity in wind tunnel

Now it will be discussed how these acoustic modes will be influenced by a uniform steady velocity in the wind tunnel. In the wave equation (4.1) a stationary reference fluid with zero velocity is assumed. The material derivative is reduced to a partial derivative with respect to the time. However, when the

reference fluid has a uniform velocity, 0Vv

, the homogeneous wave equation becomes

0''1 2

2

02

0

=−∇

∇⋅+∂∂

ppVtc

v (4.12)

In the wind tunnel the fluid moves in the x-direction, ]0,0,[0 UV =v

. Hence, the wave equation is

reduced to 0'') (/1 222 =−∇∂∂+∂∂ ppxUtco . Moreover, the acoustic pressure waves are

prescribed as harmonic modes as in (4.7) where the boundary conditions at the wall, [ yp ∂∂ /' ]y =

± L = 0 and [ zp ∂∂ /' ]z = ± L = 0, are preserved.

[ ]tixik

nzmymnxmn eekkAp mnx ωππ )zsin()ysin(Re'

2

1)-(n

2

1)-(m

++= (4.13)


34

in which L

mk my

2

π= and

L

nk nz

2

π= for integer m and n

Substituting (4.13) in (4.12) yields

( )2

02

2

2 mnxnzmymnx Mkkkkk +=++ (4.14)

In this equation 00 / ck ω= , and M is the Mach-number, 0/ cUM = . The wavenumber mnxk -

is solved from (4.14). Two values are obtained, one which corresponds to a wave propagating in

the direction of the flow and one opposite to the direction of the flow, indicated by the ± sign.

( )( )

−+−

−=± 22

2

2

002 11

1MkkkMk

Mk nzmymnx m (4.15)

Assuming a plane wave ( m = 0, n = 0) the wavenumber ±00 xk are respectively

M

kk x +

−==

++

1

2 000 λ

π and

M

kk x −

==−

−

1

2 000 λ

π (4.16)

In which the positive superscript indicates sound waves traveling in the direction of the fluid and the negative sign the opposite direction. From the result it is seen that a uniform flow will alter

the wavenumber. It can be said that the observed wavelength for a plane wave is stretched to +λ

)1( M+= λ in stream wise direction and compressed to )1( M+=− λλ in the opposite

direction. The plane waves are travelling with speed )1()( 00 McUcc +=+=+ and

)1(0 Mcc −=− in downstream and upstream direction with respect to the laboratory. This is

illustrated in figure 27. Accordingly, the frequency observed by a pressure transducer in the wall of the wind tunnel remains the same regardless of the position.

Figure 27 Doppler effect in wind tunnel

For the first non-planar wave ( m = 1, n = 0), the cut-off frequency is affected by a uniform

flow. The wavenumber 10 xk is real when ( ) 0122

1

2

0 ≥−− Mkk y , in other words, when the

frequency is larger than 2/120 )1( )4/( MLcf −> . The cut-off frequency is equal to


35

2/120 )1( 4

ML

cfc −= (4.17)

which is lowered by a factor 2/12 )1( M− . This cut-off frequency is valid at an upstream as well

as a downstream location. In spite of this, the resonance frequency of the first transversal mode does not depend on the

uniform velocity. A resonance occurs when 01 kk y = . Since the uniform flow is perpendicular

to the y-direction, the propagation speed of a wave, c , in y-direction is not altered. Therefore the resonance frequency of this mode remains the same. Hence

)4/( Lcf res = (4.18)

In this situation the wavenumber in longitudinal direction is

[ ] 0 10 =+resxk (4.19)

for a wave traveling in positive x-direction, and

[ ] )1/(2 2010 MMkk

resx −=− (4.20)

for a wave which is traveling in the opposite direction. This yields that due to a uniform velocity the resonance wave is not symmetrical in x-direction. The wave is a standing wave for positive x-direction and a propagating wave for negative x-direction. If the uniform velocity in the wind tunnel is U = 41m/sec, and the width is L2 = 0.5 m, the

cut-off frequency for the first transversal mode is cf = 342 Hz, and the resonance frequency is

resf = 344 Hz.

In conclusion, the acoustic pressure waves in the wind tunnel are a superposition of the different modes described by

( )[ ]tixik

mnx

xik

mnxnzmymn eeAeAkkp mnxmnx ωππ ) z)sin( y sin(Re'

2

1)-(n

2

1)-(m

−+ −−+ +++=

(4.21) In which the wavenumber in x-direction is

( )( )

−+−

−=± 22

2

2

002 11

1MkkkMk

Mk nzmymnx m .

Figure 28 considers four types of first transversal modes ( m = 1, n = 0). The uniform velocity, U , in the wind tunnel is set to 41 m/sec. In the first figure (a) a non-planar wave with a frequency higher than the cut-off frequency (f = 400 Hz) is shown. The wave is a propagating wave in x-direction. Several nodes and anti-nodes exist on the side walls of the wind tunnel. The nodes on the negative x-axis (upstream) follow each other up more rapidly than the nodes on the

positive x-axis (downstream). Between these anti-nodes the phase, xk x±± −= 10 10φ , increases π rad.

Besides, a nodal line is found on the middle of the tunnel. The phase of the wave in y-direction


36

is constant were at the nodal line the phase shifts π rad. The second figure (b) is a wave at the resonance frequency (f = 344 Hz). In this situation, the wave is a standing wave in the positive x-direction. The pressure displays a maximum or minimum on the side walls and is independent of the positive x position along the wind tunnel. The phase remains constant, with exception of

the nodal line in the middle of the wind tunnel were the phase shifts π rad. At negative x-direction the wave is a propagating wave with nodes and anti-nodes. Again, the phase increases

in negative x-direction and shifts π rad on the nodal line. Third, (c) a wave just above the cut-off frequency is shown (f = 342 Hz). In this situation the wave is a propagating wave in both the positive and negative x-direction. The same structure as in figure (a) is observed, although the wavelengths in x-direction are larger. Finally, the last figure (d) shows only a maximum or minimum in front of the speakers. The pressure vanishes at some distance from the sound sources. This is an evanescent transversal mode (f = 300 Hz).

(a) (b)

(c) (d)

Figure 28 Four types of modes in the wind tunnel, (a) Propagating wave, f (400Hz), (b) Resonance mode

f (344 Hz), (c) propagating wave just above the cut-off frequency f (342 Hz), (d) Evanescent mode f

(300Hz) . The uniform velocity in the wind tunnel is U = 41 m/sec. In the experiments the speakers are tuned to the first transversal resonance frequency ( f = 344

Hz), hence an acoustic field as presented in figure 28b is expected. It has to be remarked that the solution is highly idealized. No influence of damping at walls is taken into account. Furthermore, the flow is assumed uniform without any turbulence or boundary layers.


37

4.1.4 Acoustic velocity in wind tunnel with uniform flow

Once the acoustic modes have been determined for a wind tunnel with uniform flow, the acoustic velocities can be considered. The acoustic velocity in the y-direction is related to the oscillating motion of the airfoil. The acoustic velocity in y-direction, 'v where ]',','[' wvuv =

r, is related to the pressure

fluctuations by the y-component of the momentum equation,

0'

'0 =∂∂

+

∂∂

+∂∂

y

pv

xU

tρ (4.22)

The acoustic velocity for mode ( nm, ) is written in complex notation, i.e. ) ( ˆ'xkti

mnmnmnxevv

±

= mω .

Substituting this velocity and the pressure fluctuations, mnp' , given in (4.21) in the momentum

equation, results in a first order linear differential equation

( )y

pvUki mn

mnmnx ∂

∂−=± '1

'0

ρω m (4.23)

The solution of this equation is given by

( ) ( )

++

−

++=

−−−

++

−−

+tixik

mnx

myxik

mnx

my

n

nz

m

mymn

eeMkk

ikAe

Mkk

ikA

zkykc

v

mnxmnx ω

ππ

ρ

sincos

1Re'

-

0

0

2

)1(

2

)1(

00

(4.24)

Accordingly, a uniform velocity gives an extra term )/( 0 ±

mnxmy Mkkik m to the acoustic velocity

of the wave traveling in the positive respectively negative x-direction. Moreover, while the pressure shows a sinus-dependence in the y-direction, the acoustic velocity has a cosines-dependence. That is to say, when the acoustic pressure is maximal, the velocity is minimal, and vice versa. In the middle of the wind tunnel, at the location of the airfoil (y = 0), the pressure has a node and the acoustic velocity has an anti-node. The airfoil is placed at a maximum of the oscillating displacement of the flow in the experiments. The overall solution of the acoustic velocity is a superposition of (4.24) for different modes of a wave

∑∑∞

=

∞

=

=0 0

''n m

mnvv (4.25)

When only a first transversal mode ( m = 1, n = 0) is considered at a resonance frequency ( f =

344 Hz, where the width of the wind tunnel is L2 = 0.5 m), and the uniform velocity is 41

m/sec, the extra term in the acoustic velocity is equal to [ ] 1)/( 10 00 =+ +resxkMkk for the wave

in downstream direction. Accordingly, the acoustic velocity in y-direction remains the same. For

a wave in upstream direction this term is equal to [ ] )/( 10 00 =+ −resxkMkk

97.0)1/()1( 22 =+− MM , and the acoustic velocity in y-direction is slightly reduced.


38

4.2 Experiments

Now that the modes of the wind tunnel have been predicted theoretically, their structures are determined experimentally. In the first experiments the test section of the wind tunnel is separated and placed in the semi-anechoic room of the acoustic lab. The semi-anechoic room has non-reflecting walls but a hard reflecting floor. By separating the test section from the wind tunnel a pressure transducer on a translating construction can be placed inside the section and the acoustic field can be investigated. The advantage is that the acoustic field can be measured at several positions inside the section whereas in later wind tunnel experiments the pressure transducer can only be placed in the wall. The acoustic field is obtained for an empty test section and for a test section with airfoil. The second series of experiments are perfumed in the wind tunnel and are considered in the second part of this section. Again, the acoustic field is obtained for an empty test section and test section with airfoil. Moreover, the influence of a uniform velocity has been studied.

4.2.1 Semi-anechoic room

The first experiments are performed in the semi-anechoic room. In the semi-anechoic room several points in the test section are measured by placing a piezo pressure transducer (PCB) on a translating system and moving this through the wind tunnel section. This is represented in figure 29 in which the experimental setup is illustrated. The ends of the test section are open. Data acquisition is managed by a PXI-module of National Instruments; consisting of a 1.26 GHz embedded controller with widows XP, an 8-input 24 bits DSA, and a waveform generator to send a sinusoidal function to the speakers. The speakers are connected in phase, so that the acoustic waves reinforce each other. The used test airfoil is a NACA 0018 airfoil with a rectangular cavity in lengthwise direction. The width of this cavity is 40 mm. This test airfoil is clamed between the walls of the wind tunnel by a thin slice of insulation material. This isolation material also prevents a possible secondary flow through the holes in the lengthwise direction of the airfoil which may give rise to additional acoustic resonances. Later, the airfoil is fixed by means of a pin-hole-connection to the floor of the wind tunnel and by a rod with a flange at the top of the wind tunnel. The pin and rod of the test airfoil are also preventing a flow within the airfoil profile.

Figure 29 Experimental setup; 1) wind tunnel section, 2) speakers, 3) airfoil with rectangular cavity, and 4) microphone on translating system. The ends of the test section are open.


39

Empty test section

First, the resonance frequency is studied for the empty test section. Two resonances are observed in a bandwidth between 100 and 1000 Hz, one at f = 374 Hz and

the second at f = 730 Hz. The first resonance corresponds to a first transversal mode. This is

sketched in figure 30. The amplitude of the pressure oscillation decreases from the maximum at the side walls at which the speakers are placed towards the centre where a pressure node is. A phase shift π is found between the left and right side. Furthermore, the amplitude remains fairly constant in longitudinal x-direction. The second resonance is a higher order resonance with a complex structure. Therefore, it is chosen to focus in the subsequent experiments to a range between 250 and 450 Hz. Compared to the theory the resonance of the first transversal mode was expected to be found at 344 Hz. Probably the finite length of the test section can be a cause. The theory assumes an infinitely long wind tunnel and consequently no reflecting boundary conditions in the x-direction are prescribed. This explanation has to be researched in more detail, for instance by numerical simulation. Preliminary calculations by Olsman [25] using the Sysnoise code (BEM) seem to confirm this hypothesis.

Figure 30 Side view wind tunnel. The speakers at the left and right wall induce a first transversal mode. The pressure amplitude is maximal at the side walls and zero at the center.

Effect of an airfoil

Next, an airfoil is placed inside the test section and the acoustic response is measured. Surprisingly two peaks are found in the bandwidth 250 to 450 Hz, one at f = 333 Hz and the

second at f = 351 Hz. Both resonances give a first transversal mode. Remark that both

frequencies are lower than the resonance frequency of the empty test section, f = 374 Hz.

These results will be discussed later when measurements in the wind tunnel are considered. In longitudinal direction the amplitude of the pressure fluctuations decreases again. Along the wing the amplitude is almost uniform in the x-direction. Beyond the airfoil in positive x-direction the amplitude decreases monotonously.


40

4.2.2 Wind tunnel

Now, The test section is placed in the wind tunnel. Once more the acoustic field is investigated, first without airfoil, thereafter with, and finally with airfoil and a uniform velocity. The geometry of the wind tunnel, including a top view of the test section, is sketched in figure 31. One side of the test section is connected to the settling chamber and the other to a gap. Acoustic waves will reflect both at the transition from test section to settling chamber and from test section to the open gap. Hence, the test section placed in the wind tunnel can be interpreted as a pipe with finite length, similar to the experiments described above.

Figure 31 Geometry of the wind tunnel including settling chamber, test section, gap and diffuser. The top view of the test section, including the positions of the pressure transducers, is represented separately.

Empty test section

The acoustic modes in an empty test section are considered. Firstly, the resonance frequencies are determined, subsequently the structures of the acoustic modes. The resonance frequency of the empty test section is determined from the signals of two pressure transducers placed at locations I and II (see figure 31). The results are presented in figure 32 (a) for the amplitude of the oscillating pressure divided by the maximal amplitude of pressure transducers at location II, and (b) for the corresponding phase differences between

pressure transducer I and II, III φφ − . The x-axis gives the frequency in [Hz].


41

250 300 350 380 400 4500

0.2

0.4

0.6

0.8

1.0

1.2

Frequency [Hz]

|p'|

/ | p' I I

| m

ax

location I

location II358 Hz

250 300 350 380 400 450

-0.6

-0.4

-0.2

0

Frequency [Hz]

φ I -

φ II [

rad]

(a) (b)

Figure 32 Examination of the RESONANCE FREQUENCY WITHOUT AIRFOIL. The results are obtained for two different locations, I and II, in the wind tunnel, (a) amplitude divided by the amplitude of pressure transducer at location II at 358 Hz, (b) phase difference between both pressure transducers [rad].

Without airfoil the acoustical resonance peak is found at f = 358 Hz. A weaker second peak at

f = 350 Hz is also seen. This occurrence is yet not understood. Note that the frequency do not

correspond to the theoretical one, which is equal to f = 344 Hz. Again this can be due to the

fact that the test section is not infinitely long. While observing the amplitudes of the resonance frequency, something unexpected is seen. Following the theory assuming the transversal resonant mode to be dominant, it is expected that the pressure transducer closer to the side wall (location II) will display a larger acoustical pressure amplitude than the transducer closer to the centerline (location I) (figure 30). However, in figure 32a this is not true; the pressure transducer closer to the centerline gives a higher amplitude for most frequencies. First, a difference due to calibration was expected. Switching the positions of the two pressure transducers did not give any change. In addition, both pressure transducers are calibrated within 1% accuracy. The observation is therefore not caused by a calibration error. The phase difference between pressure transducer I and II is almost zero at the resonance frequency, the difference is 0.03 rad. Nevertheless, at frequencies higher than the resonance frequency the difference drops, which is unexpected since a first transversal mode comprises a standing wave in y-direction for frequencies above the cut-off frequency. A negative phase difference implies a propagating wave between the walls of the wind tunnel. To investigate the modes of the acoustic waves in more detail, the pressure is measured at five more positions denoted by A to E. First, the acoustic field is studied in upstream direction (figure 33), later at a cross section at 2.5 times the half chord length upstream, bx 5.2−= where b is the half chord length (figure 34). Figure 33 shows the acoustic field in upstream direction. Three additional pressure transducers are placed at position A, B and C (see figure 31). In the upper figure the amplitude of the acoustic waves, divided by the amplitude of pressure transducer at location A, is plotted as a function of the x-coordinate. The lower figure gives the corresponding phase differences between the measured signal and pressure transducer A in [rad]. The results are obtained for the resonance frequency f = 358 Hz.


42

-200 (C) -100 (B) 0 (A) 80 (II)

0

1/2 pi

φ -

φA [ra

d]

x-axis [mm]

-200 (C) -100 (B) 0 (A) 80 (II)0.7

0.8

0.9

1

|p'|

/ |p

' A|

Figure 33 WITHOUT TEST AIRFOIL. Examination of the acoustic field in x-direction. The x-axis corresponds to the x-axis of the wind tunnel. The speaker is placed at x = 0. The y-axis represents the amplitudes divided by the amplitude of the pressure transducer at location A (upper figure) or phase differences between the pressure transducers and transducer at location A in [rad] (lower figure). The input signal is 3V, which resulted to a pressure amplitude of 90 Pa at location A, and the frequency is set at 358 Hz.

From the figure it is seen that the amplitude decades from A. At bx 5.2−= (location C) the amplitude has dropped 27%. Also the corresponding phases are not constant along the axis of the wind tunnel, but increases. Considering the resonance frequency of the first transversal mode, these observations are not expected. Then, the acoustic field in a plane at 2.5 times the half chord length upstream, bx 5.2−= , is studied. The results are given in figure 34 in which (a) the amplitude, divided by the maximal amplitude of pressure transducer at location II, and (b) phase shift between the measured signal and the transducer at location II are given as function of different frequencies. The pressure transducers are now placed at C (top), D (right side), and E (left side) (see figure 31). When the pressure at bx 5.2−= is compared to the pressure of the transducers at location I and II, it is noted that again the pressure amplitude in front of the speakers is higher than at some distance away along the pipe. Furthermore, all the pressure transducers record the same resonance frequency, f = 358 Hz. A phase difference between C and II, and D and II of

almost 2π is observed, which is unexpected. Besides, the phase difference between I and II is expected to be zero, but a phase difference of -0.3 rad at f = 358 Hz is measured.

A remarkable observation is that around the resonance frequency the amplitude of the pressure transducer at the top C measures a higher pressure amplitude than the two pressure transducers (D and E) in the side walls. This can not be understood if it is assumed that the acoustic field is dominated by the first transversal mode. This could be due to the presence of longitudinal waves along the pipe axis (mode 0,0 , =nm ).

The phase shift between the pressure transducers at the side walls (D and E) is about π. Hence, a node exists somewhere in the middle of the wind tunnel. At the resonance frequency, f = 358


43

Hz, the amplitudes of both transducers in the side walls are roughly equal, from which can be assumed that the acoustic wave is almost symmetric at this frequency.

350 356 358 360 365

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Frequency [Hz]

|p'|

/ | p

' II| max

location II

" I

location C

" D

" E

350 356 358 360 365

-2 pi

-pi

-1/2 pi

0

Frequency [Hz]

φ -

φ II

[rad]

φ I - φ

II

φ C - φ

II

φ D - φ

II

φ E - φ

II

Figure 34 WITHOUT TEST AIRFOIL. Influence frequency on the ACOUSTIC PRESSURE close to the position of the wing (location I and II) and 2.5 times the half chord length upstream (location C, D, and E), (a) amplitude divided by the peak value of pressure transducer at location II, (b) gives the phase difference between the measured signal and input signal to the speakers in [rad].


44

Effect of an airfoil

An airfoil is now placed in the wind tunnel. This is the NACA 0018 airfoil. Again the resonance frequency is studied as well as the acoustic field at bx 5.2−= . The results for the resonance frequency are given in figure 35. The frequency response is measured by the pressure transducers at location I and II. The left figure (a) gives the amplitude of the pressure divided by the maximal amplitude of pressure transducers at location II. The figure to the right hand side (b) gives the corresponding phase differences in [rad]. This is the phase difference between the pressure transducer I and transducer II.

300 320 335 350 3750

0.2

0.4

0.6

0.8

1.0

Frequency [Hz]

|p' |

/ | p

' I I | m

ax

location I

location II

331 Hz 337 Hz

300 320 335 350 375-0.2

-0.15

-0.1

-0.05

0

0.05

Frequency [Hz]

φ I -

φ II

[r

ad

]

(a) (b)

Figure 35 Examination of the RESONANCE FREQUENCY WITH AIRFOIL, (a) amplitude divided by the maximum amplitude of pressure transducer at location II, (b) corresponding phase difference between both pressure transducers in [rad].

When the airfoil is placed in the wind tunnel the resonance frequency drops and two clear peaks in the amplitude appear, one at f = 331 Hz and the other at f = 337 Hz.

The cause of the drop in frequency, compared to the case of an empty test section, can be explained by the additional distance that a transversal wave has to travel when an airfoil is placed in the middle. At the resonance frequency the wavelength, λ , is equal to one meter when the distance between the walls is equal to L2 = 0.5 m. When the test airfoil is placed in the middle the waves locally have to travel around it and the wavelength increases which result to a lower resonance frequency. The cause of the two peaks is not understood. Probably more acoustics modes, besides the first transversal mode, arise. As the cross-section is not symmetrical anymore due to the wing, a change in these additional modes can happen. The phase difference between pressure transducers I and II, φ I - φ II, is decreasing.

Nevertheless, this increment is small, only 0.15 rad. Considering the structure of the acoustic wave at the resonance frequencies, the amplitude of the pressure transducer located closer to the speakers is this time higher than the pressure transducers located nearer to the centerline. However, the ratio of the amplitudes does not correspond to the theoretical one for a simple transversal standing wave ( m = 1, n = 0).

07.1|'|

|'|exp ==

I

II

p

pr 35.1

)sin(

)sin(

2

2 ==IL

IILtheory

y

yr

π

π


45

Where L2 is the width of the wind tunnel ( L2 = 0.5 m), and Iy and IIy are the position of the

pressure transducers at location I and II ( Iy = -0.125 m, IIy = -0.202 m).

Hereafter the acoustic field at bx 5.2−= is studied. Additional pressure transducers are placed at C (top), D (right side), and E (left side). The results are presented in figure 36 in which (a) the amplitude, divided by the maximal amplitude of pressure transducer at location II, as well as (b) the phase difference in [rad] are plotted as function of the frequency in [Hz]. The phase shift is the difference between the pressure transducers and the transducers at location II in [rad]. Several observations are made from this result. Firstly, the resonance frequency upstream remains still around 335 Hz and the two peaks of 331 Hz and 337 Hz are observed. Secondly, the amplitudes at bx 5.2−= are lower than the ones corresponding at location I and II. The amplitude dropped 58%, which is more than what was observed previously when no airfoil was placed. The amplitude decayed 13% between location II and C in the empty test section. Besides,

the phase difference between C and II, and D and II is almost 2π rad. Thirdly, the amplitudes at the side walls (D and E) are now higher compared to the amplitude at location C, which is more in agreement with the theory of a first transversal acoustical mode. Nevertheless, the amplitudes at the side walls (D and E) are not equal to each other. In addition, at the resonance frequency the phase shift is not π. This indicates an asymmetry of the acoustic field in y-direction.


46

300 320 335 350 375

0.2

0.4

0.6

0.8

1.0

Frequency [Hz]

|p'|

/ | p

' II| max

location II

" I

location E

" D

" C

300 320 335 350 375

-2 pi

- pi

- 1/2 pi

0

Frequency [Hz]

φ -

φ II

[rad]

φ I - φ

II

φ C - φ

II

φ D - φ

II

φ E - φ

II

Figure 36 TEST AIRFOIL PLACED. Influence frequency on the ACOUSTIC PRESSURE close to the position of the wing (location I and II) and 2.5 times the half chord length upstream (location C, D, and E), (a) amplitude divided by the peak value of pressure transducer at location II, (b) phase difference between the pressure transducers and the transducer at location II in [rad].


47

Effect airfoil and uniform velocity

Finally, the last experiment considers the acoustic field at bx 5.2−= when the airfoil is placed inside the wind tunnel and the wind tunnel is turned on. The uniform velocity in the wind tunnel is set to 41 m/sec. The results are presented in figure 37. The acoustic pressure is measured by pressure transducers located close to the airfoil (location I and II), and at bx 5.2−= at D (right wall), and E (left wall). The pressure transducer at location C (top) is not included. The amplitude, divided by the peak value of pressure transducer at location II, is shown in the left figure (a). The phase difference between the pressure transducers and the transducers at location II in [rad] is given in the figure to the right (b). On the x-axis the frequency is given in [Hz].

300 320 335 350 3750.2

0.4

0.6

0.8

1.0

Frequency [Hz]

|p'|

/ | p' I I

| m

ax

location II

" I

location E

" D

300 320 335 350 375

-2 pi

- pi

0

Frequency [Hz]

φ -

φ II

[rad

]

φ

I - φ

II

φ E - φ

II

φ D

- φ II

(a) (b)

Figure 37 Examination of the acoustic field WITH AIRFOIL and a UNIFORM VELOCITY in the wind tunnel. The results are acquired from the pressure transducers located at position I and II, and at

bx 5.2−= (location D and E), (a) amplitude divided by the maximum amplitude of pressure transducer at location II, (b) phase difference between the pressure transducers and transducers at location II in [rad]. The uniform velocity in the wind tunnel is 41 m/sec.

As expected from the acoustic theory, the resonance frequencies remain about the same, f =

331 Hz and f = 339 Hz. The peaks are however less sharp than without flow. It is observed

that the amplitudes of the pressure transducers at bx 5.2−= are higher compared to the results without main flow.

4.3 Transversal acoustic velocity

Although the acoustic field of the wind tunnel is not totally understood, an estimation of the acoustic velocity in y-direction, normal to the main flow, has to be made. This is considered in this section. The acoustic velocity of interest is directed in the y-direction, 'v were ]',','[' wvuv =

r. The

estimation of this transversal velocity relies on the idea to obtain the velocity at bx 5.2−= instead of directly in front of the speakers. This is done because the pressure transducers close to the speakers gave unexpected results which may be caused by evanescent modes. Close to the


48

speakers the acoustic field is complex but after the waves have traveled some distance these effects are attenuated. The acoustic velocity at bx 5.2−= and y = 0 is compared with the

acoustic velocity at the location of the airfoil, x = 0 and y = 0. To obtain this velocity the airfoil

is removed from the wind tunnel. On the flange, which normally fixes the airfoil, two pressure transducers are placed at equal distance, ε , of the midline, ε2 = 0.15 m (see figure 38).

Figure 38 Top view experimental setup for determination acoustic velocity in y-direction, 'v . The pressure

transducers are located at position III and IV , and in the middle of the side walls at bx 5.2−= at location D and E.

Firstly, the experimental transversal velocities at both positions are considered, where after the last section makes a comparison between the two.

4.3.1 Experimental determination of transversal acoustic velocity

First of all, the transversal acoustic velocity obtained experimentally at the position of the airfoil is described, thereafter at bx 5.2−= .

Acoustic velocity at the airfoil

To obtain the acoustic velocity, 'v , at the airfoil, x = 0 and y = 0, two assumptions are made.

Firstly, it is assumed that the y-velocity is uniform in the middle of the wind tunnel (this is reasonable if a fist order transversal wave is considered at the resonance). Secondly, the fluid is assumed to be inviscid. Then, the Euler equation, which describes the conservation of momentum in a fluid, may be used to express the velocity in y-direction, [ 'v ]x = 0, in terms of the pressure difference between the two pressure transducers, [ ' p∆ ]x = 0.

[ ] [ ] [ ]

ερ

2

''' 0000

=== ∆−=

∂

∂−= xxx p

y

p

Dt

vD (4.26)

The velocity and the two pressures are written in complex notation, i.e. tievv

ωˆ'= , ti

IIIIII eppωˆ' = , and ti

IVIV eppωˆ' = where the subscript III and IV refers to the locations of the

pressure transducers, III for the left side and IV for the right side. Substituting this in the Euler equation, and neglecting convective effects, results to


49

[ ]

[ ]εωρ

εωρ

ωω

0

0

00

2

ˆˆˆ

2

)ˆˆ(ˆ

i

ppv

eppevi

IIIIVx

tiIVIII

xti

−−=⇒

−−=

=

=

(4.27)

where ε2 represents the distance between the two pressure transducers in the flange on the top of the wind tunnel, ε2 = 0.15 m.

Acoustic velocity according to the pressure transducers at x = -2.5b

Next, the acoustic velocity on the cross-section at bx 5.2−= is considered. The velocity is approximated by the basic equations for harmonic waves, as described earlier. According to the theory, the amplitude of the pressure perturbations of a first transversal mode ( 0 ,1 == nm ) are equal at the side walls Ly = and Ly −= and the phase difference is π rad. In

other words, the acoustic field is anti-symmetric around y = 0. However, experiments

demonstrate that this is not always the case. The pressure transducers in the side walls (location D and E (see figure 31)) give different amplitudes in the wind tunnel experiments with (figure 34a) and without airfoil (figure 36a). Nevertheless, the phase remains almost π rad. An explanation could be that, besides the first transversal mode, also a planar wave travels along the x-axis of the wind tunnel. Hence, the wave comprises a first transversal mode ( 0 ,1 == nm )

superposed on a planar wave ( 0 ,0 == nm ). In this case, the measured amplitude of the pressure

transducers at the wall is a summation of these modes. As the amplitude at one side of the wind tunnel is positive and the other negative, a difference in amplitude is measured. This is clarified in figure 39.

Figure 39 Side view wind tunnel. The pressure wave (bold line) consists of a plane wave, 00'p , and a first

transversal mode, 10'p (with stripes). The pressure fluctuation of the plane wave is uniform in the cross

section of the wind tunnel. The pressure fluctuation of a first transversal mode is π rad shifted between the

left and right wall. Hence, the pressure transducers in the right wall, Dp' , measures a different amplitude

than the pressure transducers in the left wall, Ep' .

Accordingly, the amplitude of the planar wave, 00' p , is equal to

2

'''00

DE ppp

+= (4.28)


50

and the amplitude of the first transversal mode, 10' p , to

2

'''10

DE ppp

−= (4.29)

The planar wave travels in the x-direction. Consequently the corresponding acoustic velocity is perpendicular to the y-axis and does not contribute to the local oscillating transversal velocity,

'v . Only the amplitude of the first transversal mode ( 0 ,1 == nm ) is of interest. Hence, the

amplitude of the acoustic pressure wave at bx 5.2−= , [ ] −−=

= 10 5.210' xbxAp , is determined from

(4.29). Once this is known, the acoustic velocity, 'v 10 , can be estimated. The wind tunnel is assumed to be infinitely long, consequently reflections of waves in x-direction are ignored. Therefore the pressure transducers at bx 5.2−= are assumed to record only the

pressure waves in negative x-direction, [ ] [ ])(

5.2 Re'xkti

xbxxmnepp−−+−

−= = ω . In addition, a first

harmonic mode is considered, that is to say m = 1 and n = 0, and consequently Lk y 2/ 1 π= ,

and 00 =zk . Moreover, the acoustic velocity at (x,y) = (0,0) is of interest. The acoustic velocity

described earlier in (4.24) reduces to

[ ]

+=

+=

−−

+−−

=

=

1Re

)cos(

1Re'

10

10 0

1

00

)(10

10 0

1

1

000

010

-10

ti

x

x

y

xkti

x

x

y

yy

x

eAMkk

ik

c

eAMkk

ikyk

cv x

ω

ω

ρ

ρ (4.30)

4.3.2 Comparison transversal acoustic velocities

Both velocities are compared to each other to verify if the transversal acoustic velocity at two and a half times the half chord length is suitable to determine the transversal acoustic velocity at the airfoil. The results are presented in figure 40a to 40d. The uniform velocity is zero at the both upper figures (a) and (b), while in the lower figures (c) and (d) the uniform velocity is equal to U = 41 m/sec. The amplitude of the acoustic velocity, divided by the amplitude estimated at x = 0 and at 358 Hz, is shown in figures to the left, the corresponding phase in [rad] in the figures to the right.


51

350 354 358 362 3660.2

0.4

0.6

0.8

1

Frequency [Hz]

| v' |

/ | v

' x=0| f=

35

8H

z

Uniform velocity = 0 [m/sec]

v' at airfoil

v' estimatedat x = -2.5b

350 355 360 3650.5

1

1.5

2

Frequency [Hz]

phase

v'

[rad]


v' at airfoil


(a) (b)

350 354 358 362 3660.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

| v' |

/ | v

' x=0| f=

35

8H

z


v' at airfoil


350 355 360 3650.5

1

1.5

2

Frequency [Hz]

phase

v'

[rad]


v' at airfoil


(c) (d) Figure 40 Study of the ACOUSTIC VELOCTIY in y-direction calculated at the location of the airfoil ‘’, or at 2.5 times the half chord length upstream “”. Figure (a) and (b) uniform velocity is zero, U = 0, (c)

and (d) uniform velocity is set at U = 41 m/sec. The amplitude, divided by the amplitude calculated at

x = 0 and at f = 358Hz, is given in the figures to the left, the corresponding phase in the figures to the

right. The test section is empty.

The estimated acoustic velocities give a rather good correspondence. When no uniform velocity is set, the averaged error is 1.3% in a bandwidth around the resonance frequency, 356 Hz ≤ f

≤ 360 Hz. For a uniform velocity of U = 41 m/sec the average error is 2.6% in the same bandwidth. However, the phase shift is consistent lower. A difference of 0.15 rad is found, which corresponds to 8.5 degrees. This is ignored in the further experiments. It will further be assumed that the velocity 'v measured by using the pressure transducers in the side wall at bx 5.2−= remains valid in the presence of the airfoil.

4.3.3 Conclusion

In conclusion, the experiments of the acoustic field show unexpected results which do not correspond to the theoretical single transversal resonant mode. Probably the acoustic field is not dominated by this mode. However, a reasonable estimation can be made of the acoustic velocity in the y-direction at the airfoil position. More complicated effects require consideration in the acoustic theoretical model, like wall admittance, vibrations of the wall which may couple to the pressure and velocity fluctuations. In


52

addition, the shape of the wind tunnel may influence the acoustic field. The finite length of the test section may alter the cut-off frequency. Numerical calculations with Sysnoise by Olsman [25] seem to confirm that the finite length of the test section does affect the acoustic field. The finite length of the section at least increases the resonance frequency compared to the first transversal resonance of an infinite long channel. Furthermore, the speakers do not radiate uniform acoustic waves, that is to say, in the near field evanescent modes may contribute significant. More study by numerical calculation is required to explain the measured results.


53

Chapter 5

Dynamic response of NACA 0018 airfoil The dynamic response of the NACA 0018 is compared to the thin airfoil model in this chapter. In this way the measurement method can be verified. The dynamic response is expressed in a local pressure difference across the airfoil. This pressure difference results to net aerodynamic forces and allows prediction of the performance of the airfoil compared to other airfoils. How the comparison between the NACA 0018 and thin airfoil model is made, is put forward in the first section. The experimental results are discussed in the second section. Here, the influences of the Strouhal number, the amplitude of the acoustic pressure waves, and the angle-of-attack are investigated.

5.1 Experimental parameters

The pressure difference over the airfoil is adapted to a dimensionless form to make easy comparison. This is considered firstly. Subsequently, the parameters are gained from the experiments.

5.1.1 Dimensionless pressure difference

The stationary pressure difference is made dimensionless, hereafter the oscillating one. A stationary pressure difference over the airfoil is induced by the angle-of-attack, α . The local theoretical pressure difference between the top- and bottom sides of the airfoil,

bottomtop ppp −=∆ , is given by

( )2/tan2 20 θαρ Up =∆ (5.1)

Where 0ρ is the density of the fluid, α the angle-of-attack, U the main fluid velocity, and θ the

angular coordinate to describe a local position on the airfoil. The pressure difference is expressed

in a dimensionless form by dividing it through the dynamic pressure, 202

1 UP ρ= where 0ρ is

the density and U the main velocity of the fluid. The stationary local pressure coefficient, pc ,

is defined by

2

021 U

pc p ρ

∆= (5.2)


54

The non-stationair pressure distribution will now be considered. The oscillating pressure

distribution, tiepp

ω ˆ'= , is given by

( ))sin( )2/tan()( ˆˆ0

200 θθρφρ StiStCyStiUp +=−= (5.3)

Where tie

ωφφ = represents the acceleration potential (see section 2.3 Oscillating airfoil), i the

imaginary number, i.e. i2 = -1, St the Strouhal number which is defined as the product of half

chord length and frequency of the oscillations divided by the uniform velocity; UbSt ω= , and

0y the dimensionless displacement of the airfoil, byy /ˆ0 = . )(StC is the Theodorsen’s

function, which contains the zero order, )(0 iStK , and first order, )(1 iStK , of the modified

Bessel equation of the second kind, )()(/)()( 011 iStKiStKiStKStC += (Appendix D). From

the equation it is seen that the oscillating pressure distribution contains a " " 0ySti - part. This

part can be rewritten as a function of the acoustic velocity, ti

airfoil evvω

ˆ' = , and main velocity,

U .

U

v

b

y

U

biyiSt

airfoilˆˆ

0 ==ω

(5.4)

Here tie ˆˆ'

' ωωω airfoilti

airfoil veyiDt

Dyv === is independent of the x-coordinate. Nevertheless, a

transversal acoustic velocity will be acquired from the experiments instead of the velocity of the airfoil. An upwards displacement of the airfoil corresponds to a downwards motion of the acoustic velocity, i.e. the acoustic velocity has an opposite sign compared to the velocity of the

airfoil, airfoilfluid vv '' −= . Then, the pressure distribution can be written as

( ))sin( )2/tan()(ˆˆ0 θθρ StiStCvUp fluid +−= (5.5)

Consequently, the local pressure difference, bottomtop ppp ˆˆˆ −=∆ , for a symmetrical airfoil is equal

to pp ˆ2ˆ −=∆ . This pressure difference is divided by the dynamic pressure, 22/1 UP ρ= , to

make it dimensionless.

[ ])sin()2/tan()(ˆ

4ˆ

202

1θθ

ρiStStC

U

v

U

p fluid +

=

∆ (5.6)

Or for the local oscillating pressure coefficient, pc

[ ])sin()2/tan()(4ˆ

ˆˆ

021

θθρ

iStStCvU

pc

fluid

p +=∆

= (5.7)

The formula given in (5.7) is used to compare the experiments of the oscillating pressure difference to the thin airfoil model. From this moment on the acoustic velocity of the fluid,

fluidv is presented by v .


55

5.1.2 Experimental variables

Next, the experimental variables in the thin airfoil model are obtained from the experiments.

These comprise stationary parameters, like the angle-of-attack α , angular coordinate θ, uniform velocity U , Strouhal number St , and dynamic parameters like the oscillating velocity 'v , and

oscillating pressure difference over the test airfoil, p∆ . Besides, in some experiments the

amplitude of the acoustic pressure, | p ’|, is needed. The parameters are discussed in successive

order.

αααα

The airfoil can be rotated in the wind tunnel in order to select an angle-of-attack, α . This angle is fixed.

θθθθ

In the analytic model (chapter 2, oscillating airfoil) a thin airfoil is mapped to a circle in the ζ-plane, which has its midpoint located at the origin (see figure 41). On the flat plate, the pressure transducers are located at 17 % of the chord length (c = 0.165 m) from the leading edge, or d = 0.055 m from the midpoint. When this is transformed to the ζ-

plane, the angular coordinate becomes =−= )/(acos bdθ θ = 2,301 rad, where b is the

half chord length.

Figure 41 Conformal mapping method U and St

The pressure coefficient depends on the flying speed, U , and the Strouhal number, St . The flying speed is equal to the uniform velocity in the wind tunnel. The frequency is fixed at the first transversal resonance frequency of the wind tunnel, which is f = 337

Hz ( f 2πω = ). As a result, the Strouhal number is also known, UbSt ω= .


56

Theodorson’s function, C(St)

The local pressure difference includes the Theodorson’s function, )(StC . Its numerical

value can be calculated by using the zero- and first order modified Bessel functions;

)()(/)()( 011 iStKiStKiStKStC += (Appendix D).

v

The transversal acoustic velocity is investigated by the acoustic field and can be determined from wall pressure measurements. The method is discussed previously (section 4.3 Acoustic velocity) and will now be applied.

∆∆∆∆ p

As said before, the experimental local pressure difference is the difference between the pressure at the top- and bottom sides of the airfoil. To investigate the stationary pressure difference the averaged value is taken. To extract the oscillating local pressure difference, which corresponds to the oscillating motion, the lock-in method is used. The driving signal to the amplifier of the speakers is taken as reference signal.

| p ’|

The amplitude of the acoustic pressure is acquired from the pressure transducers at

bx 5.2−= . The amplitude is equal to one half times the difference in amplitude of the pressure transducer on the left and right side walls of the wind tunnel (section 4.3 Acoustic velocity). As the amplitude varies with the main velocity, an averaged value is taken.

Now that all the parameters are known a comparison between the thin airfoil model and the experimental pressure coefficient of the NACA 0018 can be made.

5.2 Comparison NACA 0018 airfoil to Thin airfoil model

The next section gives the comparison between the experimental pressure coefficient of the NACA 0018 airfoil and the coefficient according to the thin airfoil model. In the first situation, the stationary dynamical behavior will be compared, thereafter the oscillating one. In this case, the influences of the Strouhal number, the strength of the acoustic perturbations, and the angle-of-attack are discussed in sequence.

5.2.1 Stationary local pressure coefficient

The stationary local pressure coefficient is examined. The angle-of-attack is the only parameter which is varied.

Influence angle-of-attack

To study the influence of the angle-of-attack on the stationary local pressure coefficient, the

coefficient, )/(2

021 Upc p ρ∆= , is plotted as function of the angle-of attack in figure 42. The

experiments are performed for two different main velocities, U = 22 m/sec and U = 41 m/sec. The solid line represents the experimental stationary pressure coefficient, the dotted line the one


57

predicted by the thin airfoil theory, and the dash-dotted line gives the results of a numerical simulation (Panel method), which is done by Olsman [26].

-15 -10 -5 0 5 10 15-4

-3

-2

-1

0

1

2

3

4

angle-of-attack [degrees]

cp

Stationary local pressure coefficient

U = 22 [m/sec]

U = 41 [m/sec]

Thin airfoil theory

Panel method

Figure 42 Comparison EXPERIMENTAL AND THEORETICAL STATIONARY local pressure

coefficient, pc , as function of angle-of-attack, α , for NACA 0018. The experimental coefficients are

acquired for two different velocities, U = 22 [m/sec] and U = 41 [m/sec]. The theoretical coefficients according to the thin airfoil theory [5] and to the numerical simulation (Panel method) [26] are given both.

The result shows a characteristic plot for the local stationary pressure coefficient. The coefficient depends linear on the angle-of-attack for angles between α = -10 and α = 10 degrees. In this

region the slope, αddc p / , is equal to 12.5 [1/rad]. The theory however predicts a change of

αddc p / = 8.9 [1/rad]. This difference can be explained by the theory describing the pressure

difference for an infinitely thin plane, while the experimental airfoil has a finite thickness. Numerical simulation (Panel method) by Olsman [26] confirms this hypothesis. The results show a slope of 1.5 times higher for the NACA 0018 airfoil compared to the thin airfoil.

At α = ±13° the local stationary pressure coefficient becomes maximal or minimal. When the angle is further increased the coefficient suddenly drops. This phenomenon is called stall. At this point the boundary layer separates from the airfoil and a large wake is formed. Note that at α = 0 the stationary pressure coefficient is not exactly zero, as would be expected from the theory. Probably this deviation is a result of the finite accuracy of which the experimental wing is placed in the wind tunnel and due to an inexactness of the rotation of the angle-of-attack (0.5 degrees).


58

5.2.2 Oscillating local pressure coefficient

This section considers the oscillating local pressure coefficient. First, the thin airfoil model is compared to the experimental one for different Strouhal numbers. To investigate the dynamic behavior of the airfoil in more detail two parameters are varied, the amplitude of the acoustic perturbations and the angle-of-attack.

Influence Strouhal number

In this experiment the influence of the Strouhal number on the oscillating pressure coefficient is examined. The pressure differences across the NACA 0018 are compared to the thin airfoil model for several Strouhal numbers, a Strouhal number of St = 10.5 corresponds to a low wind velocity (U = ~17 m/sec), and a Strouhal number of St = 3.1 to a high velocity (U = ~57 m/sec). The results are presented in figure 43 (a) for the amplitudes and (b) for the phases. The experimental

oscillating pressure coefficient, )ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= , is the solid line, the theoretical one,

[ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += , the dotted line.

First, the behavior determined by the thin airfoil model is considered. The amplitude of the theoretical oscillating pressure coefficient shows a straight line when the Strouhal number exceeds St = 3. At this point the absolute value of the Theodorson’s function, )(StC , tends to

become a constant. Consequently, the coefficient depends linearly on the Strouhal number with

a slope of )sin(4 θ = 2.98. In addition the phase is ½ π. For these high numbers the oscillating

pressure difference is dominated by inertial forces corresponding to the apparent mass of the airfoil. In the situation that the Strouhal number is lower than St =3, the oscillating pressure coefficient is a non-linear function of the Strouhal number. A minimum around St = 0.8 is found. In this region the pressure difference due to bound vorticity is dominating. The experimental oscillating pressure coefficients are acquired for Strouhal numbers ranging between 3 and 12. The absolute value is consistent higher, 34% to 40%, compared to the theoretical one. Nevertheless a straight line is found. The experimental phase agrees well to the thin airfoil model, 0.5% to 6.4%. An explanation of the deviation of absolute value could be the shape of the airfoil (finite thickness). Preliminary numerical calculations by Olsman [26] confirm this assumption as they accurately reproduce the experimental data. However, another uncertainty in the calculation is the air displacement at the airfoil. Since the acoustic wave may not be a pure transversal wave, the acoustic velocity at the airfoil may be different. This could result in a major difference. Data signals give additional information about the dynamic pressure on the airfoil. Details of the data signal of the pressure transducer at the top of the airfoil are presented in Appendix J. The response signal of the pressure transducer is a pure sinusoidal function for high Strouhal numbers ( St = 4.9 to St = 11.0). For lower Strouhal number St < 4.9 (increased velocity), more noise is observed in the signal.


59

Figure 43 Local OSCILLATING pressure coefficients, )ˆ /(ˆˆ 021 vUpc p ρ∆= , obtained on NACA 0018

airfoil are compared to the thin airfoil model given by F.C. Fung, )]sin( )2/tan()([ 4ˆ θθ StiStCc p += .

The pressure coefficient is given as function of the Strouhal number. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 126 Pa. Figure (a) gives the absolute

values, (b) the corresponding phase in [rad].


60

Influence amplitude acoustic perturbation

To investigate the dynamics of the NACA 0018 more, the amplitude of the acoustic perturbation is varied. The amplitude of driving signal to the amplifier of the speakers is changed and the corresponding oscillating pressure coefficient is obtained. The results are presented in figure 44 (a) for the absolute values and (b) for the phases. The x-axis corresponds to the Strouhal number, the y-axis to the pressure coefficient according to the thin airfoil model,

[ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += (dotted line) and to the experimental ones,

)ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= (solid lines). The averaged pressure fluctuation ranges from 49 Pa to

166 Pa. From the results it can be found that the influence of the strength of the acoustic perturbations

on the oscillating pressure coefficient is minimal. The pressure difference over the airfoil, p∆ , is

proportional to the transversal oscillation, v , and since the acoustic perturbation directly influences the transversal oscillation it can be concluded that the pressure difference is proportional to acoustic perturbations as well. The phase displays a maximal deviation of 0.11 rad (6 degrees).


In addition, the influence of the angle-of-attack on the oscillating dynamical behavior of the airfoil is examined. The coefficient is acquired at several angles-of-attack, α . The angles are varied between α = 0 and α = 12.5 degrees. As the airfoil is symmetric, only positive angles-of-attack are taken. Figure 45 (a) gives the results for the absolute values and (b) for the phases. The x-axis corresponds to the Strouhal number, the y-axis to the theoretical oscillating pressure coefficient,

[ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += (dotted line) and to the experimental ones,

)ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= (solid lines).

For low angles-of-attack the oscillating pressure coefficient does not change much, i.e. the lines for α = 0°, α = 2.5°, and α = 5° are almost identical. When the angle is enlarged the experimental coefficient deviates more, particular for high Strouhal numbers. At Strouhal number St = 11 the coefficient of α = 12.5° is 10.3 % lower compared to α = 2.5°. Since in this case the main velocity is not that high an explanation could be that the boundary layers at the airfoil are bigger and that flow separation will occur faster. This results in a lower pressure difference. When the main velocity is increased the Reynolds number becomes larger and flow separation occurs at a later stage. In these points the experimental oscillating pressure coefficients are kept to the lines of small angles-of-attack. The experimental phase remains around 1.3 rad. On the basis of thin airfoil theory no influence of the angle-of-attack on the pure transversal oscillating airfoil is expected, that is to say, the oscillating pressure coefficient is independent on the angle-of-attack, α . This parameter only induces a stationary pressure difference.


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Figure 44 Influence of the amplitude of the ACOUSTIC PERTURBATION on the local

OSCILLATING pressure coefficient, )ˆ /(ˆˆ 021 vUpc p ρ∆= . The solid lines represent the experimental

values, the dotted line the coefficients according to the thin airfoil model. The x-axis gives the Strouhal numbers, St . The frequency of the speakers is set at 337 Hz. Figure (a) absolute values, (b) corresponding phases in [rad].


62

Figure 45 Influence of ANGLE-OF-ATTACK (α in [degrees]), on the local OSCILLATING pressure

coefficient, )ˆ /(ˆˆ 021 vUpc p ρ∆= . The solid lines represent the experimental values, the dotted line the

coefficients according to the thin airfoil model. On the x-axis the Strouhal numbers, St , are given. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’|~ 100

Pa. Figure (a) the absolute values, (b) corresponding phases in [rad].


63

5.3 Conclusion

In conclusion, to validate the measurement method the dynamical behavior of the NACA 0018 airfoil is compared to the thin airfoil model. The measurement method is proved suitable for determination of the static and oscillating dynamical behavior of the airfoil. The local stationary pressure coefficient is measured for several angles-of-attack. The

experimental slope of the stationary pressure coefficient, αdcd p /|| , is 50% higher than the

predicted one by the model. This is explained by the finite thickness of the experimental airfoil. The local oscillating pressure coefficient is expressed as a function of the Strouhal number,

UbSt / ω= . The oscillating pressure coefficients of the NACA 0018 for a zero angle-of-attack gave consistently higher amplitude, 34% to 40%, compared to the thin airfoil model. This deviation is also expected to be related to the finite thickness of the wing. The phase displays a deviation between 0.5% and 6.4 % from the thin airfoil model. Furthermore, the oscillating pressure coefficient does not depend on the amplitude of the acoustic pressure perturbations. Moreover, a small angle-of-attack does not affect the coefficient. In spite of this, if the angle-of-attack becomes 10 degrees or larger, a drop in oscillating pressure coefficient of 10% is observed at Strouhal number St = 10.9.


64


65

Chapter 6

Influence of a cavity A cavity is made in a NACA 0018 airfoil and the impact of this cavity on the dynamical behavior is studied. Firstly, the stationary local pressure coefficients are considered, hereafter the oscillating pressure coefficients. For the latter one, the influence of the cavity on the coefficient is investigated for several Strouhal numbers, amplitude of acoustic perturbations, and angles-of-attack.

6.1 Experimental comparison airfoil with cavity and NACA 0018

6.1.1 Stationary local pressure coefficient

The local stationary pressure coefficient depends on the angles-of-attack, α . Since the airfoil with cavity is not symmetrical, a definition of the positive and negative angle is required. The location of the cavity is defined on the top side of the airfoil. A positive angle-of-attack is taken as the leading edge of the airfoil going upward, the negative angle-of-attack as the leading edge going downward. This is represented in figure 46.

Figure 46 Definition positive and negative angle-of-attack


In this experiment the effect of the angle-of-attack on the steady pressure coefficient is examined. The stationary pressure coefficient has been measured for several positive and negative angles-

of-attack. The results are presented in figure 47. The coefficient, )/(2

21 Upc p ρ∆= where

bottomtop ppp −=∆ , is plotted as function of the angle-of-attack, α , in [degrees]. The NACA

0018 airfoil and airfoil with cavity are compared to each other (solid lines). The stationary pressure coefficient according to the thin airfoil theory is plotted as well (dotted line). The main velocity is equal to U = 41 [m/sec]. A cavity clearly changes the stationary pressure coefficient of the NACA 0018 airfoil. The coefficient is not anti-symmetrical anymore. The coefficient is roughly unchanged for positive


66

angles, whereas at negative angles the stationary pressure coefficient for the airfoil with cavity increases by a factor of 1.5. Note that at a zero angle-of-attack, α = 0, a negative coefficient is observed. A cavity on the top side of the airfoil deteriorates the static dynamical behavior of the airfoil.

Despite of the non-symmetrical characteristics, the stall angles are unchanged, i.e. α = ± 13°.

-20 -15 -10 -5 0 5 10 15 20-4

-3

-2

-1

0

1

2

3

4

angle-of-attack [degrees]

cp

Comparison NACA 0018 and airfoil with cavity

NACA0018 with cavity

NACA0018

Thin airfoil theory

Figure 47 Comparison local STATIONARY PRESSURE COEFFICIENTS, pc , between the NACA

0018 airfoil and the airfoil with cavity (solid lines), and the pressure coefficient according to the thin airfoil theory (dotted line). The steady coefficients are plotted as function of the angle-of-attack, α , in degrees.

The velocity is equal to U = 41 [m/sec].

6.1.2 Oscillating local pressure coefficient

After the stationary behavior of the airfoil is considered, the influence of the cavity on the oscillating behavior is examined. Three different experiments are performed. The first experiment studies the oscillating pressure coefficient as function of the Strouhal number. In addition, the pressure in the cavity is examined. The second and third experiments study the oscillating pressure coefficient for different amplitudes of acoustic pressure waves, and positive- and negative angles-of-attack.

Influence Strouhal number

In this experiment the influence of the Strouhal number on the oscillating pressure coefficient is examined.

The local experimental oscillating pressure coefficient, )ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= , of a NACA

0018 and the airfoil with cavity (solid lines), as well as the local oscillating pressure coefficient

according to the thin airfoil theory, [ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += (dotted line), are

plotted as function of the Strouhal number. Figure 48 (a) gives the absolute values and (b) the


67

corresponding phase in [rad]. On the x-axis the Strouhal numbers are plotted, St = 3.06 corresponds to a high velocity (U = 57 m/sec), and St = 10.53 to low main velocity (U = 17 m/sec).

Figure 48 Local OSCILLATING pressure coefficient, )ˆ /(ˆˆ 021 vUpc p ρ∆= , obtained on the NACA 0018

and on the airfoil with cavity are compared to each other (solid lines). The oscillating pressure coefficient

according to the thin airfoil model, [ ])sin( )2/tan()(4ˆ θθ StiStCc p += , is given by the dotted line. The

coefficients are given as a function of the Strouhal number, St. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 126 Pa. Figure (a) gives the absolute values,

(b) the corresponding phase in [rad].


68

Comparing the results of the NACA 0018 airfoil to the airfoil with cavity, two deviations are noticed. Firstly, the local oscillating pressure coefficient is increased at St = 3.5. While the experimental and theoretical coefficients of the NACA 0018 decreases for low Strouhal numbers, the cavity causes a peak in oscillating pressure coefficient. Apparently, the dynamical behavior of the airfoil with cavity changes for a wind velocity above 36 m/sec ( St = 4.9). Secondly, the local oscillating pressure coefficient for the airfoil with cavity has a different slope for Strouhal numbers larger than St = 4.9. For an airfoil with cavity the slope is equal to

dStcd p /|ˆ| = 4.5, whereas the slope for the normal airfoil is dStcd p /|ˆ| = 3.9.

The phase is somewhat larger for the airfoil with cavity than without. The deviation between the two airfoils is averaged 0.11 rad (6.30 degrees). The deviation of the oscillating pressure coefficient of the NACA 0018 and airfoil with cavity is given as function of the Strouhal number. This is represented in figure 49. In this figure only the absolute values are considered.

Figure 49 Difference between the non-steady pressure coefficient of the NACA 0018 and the airfoil with

cavity, | 0018 ˆˆ

NACApcavityp cc − |, as function of the Strouhal number, St . The frequency of the speakers is

set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 126 Pa.

Signal in cavity

The difference between the pressure in the cavity and on the top side of the airfoil may give information about the flow in the cavity. This experiment, for this reason, investigates the pressure in the cavity for different Strouhal numbers. The experimental values are compared to theoretical values. These are determined by means of the thin airfoil model, were the local pressure at the cavity is transformed to the top side of the flat plate as given in figure 50. Hence, the theoretical value assumes the pressure difference at the top side when no cavity is included.

Figure 50 Transformation pressure in cavity to thin airfoil


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The results of the experiments are presented in figure 51. The dimensionless pressure difference

between the cavity and top side, )ˆ/()ˆˆ( 021 vUpp cavitytop ρ− , is plotted as function of the Strouhal

number. The amplitudes are given in figure (a) and the corresponding phase in [rad] in (b).

Figure 51 Dimensionless pressure differences between the top side of the airfoil and the CAVITY, ( p top

– p cavity) / )ˆ ( 021 vUρ , as function of the Strouhal number, St . The experimental values are given by

the solid line, the theoretical by the dotted line. The frequency of the speakers is 337 Hz, and the amplitude of the acoustic pressure is | p ’|~ 126 Pa. Figure (a) gives the absolute values, (b) the

corresponding phase in [rad].


70

The absolute values of the dimensional pressure difference have similarities to the difference between the top- and bottom side of the airfoil. At Strouhal number St = 3.5 a peak in pressure

difference is seen. Besides, for Strouhal numbers larger than St > 4.9 the slope dStcd p /|ˆ| is

1.8 times larger than the theory. The experimental phase is higher, 0.9 to 0.2 rad, compared to the theoretical phase. Note that at St = 11 the phase suddenly drops. For decreasing Strouhal numbers, the data signal of the pressure transducers in the cavity has increasingly more noise compared to the data signal at the top side of the airfoil. (Appendix J).

Influence amplitude acoustic perturbation

To investigate the dynamics of the airfoil with cavity in more detail, the amplitude of the acoustic perturbation is varied. The oscillating pressure coefficient is obtained for several amplitudes of the acoustic perturbations. The results are presented in figure 52. The local experimental oscillating pressure

coefficient, )ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= , for NACA 0018 airfoil with cavity is represented by the

solid lines, the theoretical pressure coefficient, [ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += , is

given by the dotted line. Figure 52 (a) gives the absolute values and (b) the corresponding phases in [rad]. On the x-axis the Strouhal numbers are plotted. The amplitude of the pressure fluctuation ranges from 49 Pa to 166 Pa. As expected from the results of the original NACA 0018 airfoil, the influence of the amplitude of the acoustic perturbations on the local oscillating pressure coefficient is small, i.e. the maximal deviation is 8.7%. A slight shift of the peak of the oscillating coefficient is noticed for | 'p | =

54 Pa. The peak is lower and starts already at St = 4.9. A deviation is seen in the phase for Strouhal numbers lower than St < 3.5, which corresponds to the peak in oscillating pressure coefficient. The lower the perturbation, the more the phase suddenly increases. This was not observed in the previous experiment on the NACA 0018.


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Figure 52 Influence of the amplitude of the ACOUSTIC PERTURBATION on the local

OSCILLATING pressure coefficient, )ˆ /(ˆˆ 021 vUpc p ρ∆= , on the AIRFOIL WITH CAVITY (solid

lines). The oscillating pressure coefficients according to the thin airfoil theory are given by the dotted line. The coefficients are given as function of the Strouhal number. The frequency of the speakers is set at 337 Hz. Figure (a) gives the absolute values, (b) gives the corresponding phases in [rad].


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The influence of the angle-of-attack on the dynamical behavior of the airfoil with cavity is studied. Since the airfoil with cavity is not symmetric, the results for positive angles and negative angles-of-attack are discussed separately. Positive angle-of-attack The influence of positive angles-of-attack on the local oscillating pressure coefficient is considered in figure 53. The experimental oscillating pressure coefficient,

)ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= , for the airfoil with cavity is represented by the solid lines, the

theoretical one, [ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += , is given by the dotted line. Figure (a)

gives the absolute values and (b) the corresponding phases in [rad]. On the x-axis the Strouhal numbers are plotted. The angle-of-attack is varied between α = 0° and α = +10° For low angles-of-attack the local oscillating pressure coefficient of the NACA 0018 airfoil with cavity is the same as for the original NACA 0018 profile. The lines of α = +2.5° and α = 0° remain close to each other. Unexpected is the drop in coefficient around St = 4.3 for an angle-of-attack of α = +5 degrees. Furthermore, at an angle-of-attack of α = + 10° the peak at St = 3.5 is attenuated and the oscillating pressure coefficient for Strouhal numbers larger than St > 6.0 are 13% lower.


73

Figure 53 Influence of POSITIVE ANGLE-OF-ATTACK ( α+ ) on the OSCILLATING pressure

coefficient of the airfoil with cavity, )ˆ /(ˆˆ 021 vUpc p ρ∆= . The coefficients are acquired for several

Strouhal numbers, St . The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’|~ 97 Pa. Figure (a) absolute values, (b) corresponding phases in [rad].


74

The difference between the NACA 0018 and the airfoil with cavity is considered for high positive angles-of-attack. A high angle-of-attack will induce flow separation and subsequently a large wake will be formed which in turn affects the pressure difference across the airfoil. In figure 54 the amplitude of the airfoil with cavity is compared to the NACA 0018 airfoil. The absolute value of (a) the oscillating pressure coefficient at α = +5 degrees is plotted in the left figure, (b) the coefficient at α = + 10 degrees in figure to the right. On the x-axis the Strouhal number is given.

(a) (b)

Figure 54 Comparison absolute values of oscillating pressure coefficient, | pc |, between the NACA 0018

and airfoil with cavity, (a) for an angle-of-attack of α = + 5 degrees, (b) for α = + 10 degrees. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 97 Pa.

The maximal deviation between the airfoils at α = +5° is 12.6% (around St = 3.5). At α = + 10° the airfoil with cavity remains high for Strouhal number St = 10.9, whereas the NACA 0018 dropped (deviation is 24%). Negative angle-of-attack Hereafter, the influence of negative angles-of-attack on the oscillating behavior of the airfoil with cavity is considered. The local experimental coefficients over the airfoil with cavity,

)ˆ /(ˆˆ02

1exp_

vUpc p ρ∆= , are presented in figure 55 (solid lines). The theoretical oscillating

pressure coefficient, [ ])sin( )2/tan()(4ˆ_

θθ StiStCctheoryp += , is given by the dotted line.

Figure 55 (a) gives the amplitudes and (b) the corresponding phases in [rad]. The x-axis gives the Strouhal numbers. The angle-of-attack is varied between α = 0° and α = -10°. The deviation for small negative angles-of-attack is averaged 6.7%, with the exception at St = 3.5 were the difference is 11.4 %. However, at this Strouhal number the peak in oscillating pressure coefficient is attenuated continuously for increasing angles-of-attack. For an angle-of-attack of α = -10 degrees the peak is not found, although another peak is observed around St = 5. The phases for high Strouhal numbers ( St > 6) are averaged 0.10 rad higher compared to the phases at zero angle-of-attack.


75

Figure 55 Influence of NEGATIVE ANGLE-OF-ATTACK (-α ) on the OSCILLATING pressure

coefficient of AIRFOIL WITH CAVITY, )ˆ /(ˆˆ 021 vUpc p ρ∆= . The coefficients are acquired for several

Strouhal numbers, St . The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 97 Pa. Figure (a) absolute values, (b) corresponding phases in [rad].


76

The amplitudes of the oscillating pressure coefficient of the NACA 0018 and the airfoil with cavity are compared in figure 56. The absolute value of the coefficients at α = -5° are plotted in the left figure (a), the coefficients at α = - 10° in figure to the right (b). On the x-axis the Strouhal number is given. Large deviations are observed for the oscillating pressure coefficient at α = -10°. Again the coefficient at St = 10.9 is 26% higher for the airfoil with cavity.

(a) (b)

Figure 56 Comparison absolute values of the oscillating pressure coefficient, | pc |, between the NACA

0018 airfoil and the airfoil with cavity, (a) for an angle-of-attack of α = - 5 degrees, (b) for α = - 10 degrees. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 97 Pa.

6.2 Conclusion

In conclusion, a cavity in the airfoil influences the local stationary and oscillating dynamical behavior of the airfoil. The stationary pressure coefficient over the airfoil depends strongly on the sign of angle-of-attack, α . The coefficients are 1.5 times larger for negative angles-of-attack. The oscillating dynamical behavior of the airfoil with cavity has a peak in coefficient around St = 3.5. Furthermore, if the Strouhal number is larger than St > 4.9 the slope is higher compared to the NACA 0018 airfoil. Varying the strength of the acoustic perturbations does not influence the absolute value of the oscillating pressure coefficient, however the phases for St < 3.5 increases when the amplitude of the acoustic perturbations decreases. The oscillating pressure coefficient is also not influenced for low angles-of-attack. Nevertheless, if the angles-of-attack becomes

larger than α = ± 5 degrees, a deviation in dynamical behavior is noticed. At α = + 5 degrees an unexpected decrement in coefficient is found around St = 4.3. At α = + 10 degrees the peak at St = 3.5 is attenuated and the coefficient around St = 10.9 remains high whereas the oscillating pressure coefficient for the NACA 0018 dropped. In this point the coefficient is 24% higher. At a negative angle-of-attack of α = - 10 degrees the peak of oscillating pressure coefficient is found around St = 4.9 instead of St = 3.5.


77

Conclusions and recommendations As final part of this thesis conclusions are draw and recommendations are presented

7.1 Conclusions

Objective

Whitin the framework of the European Project VortexCell 2050 it will be investigated whether an airfoil with cavity is an appropriate design for a HALE aircraft. The dynamical behavior of the new airfoil therefore needs to be assessed. This includes determination of non-stationary forces due to vibrations. This thesis is a preliminary study to determine experimentally the static and oscillating dynamical behavior of an airfoil with cavity. For this, a new experimental method is verified to obtain the dynamical behavior.

Principles of the measurement method

The dynamical behavior of an airfoil is measured in the wind tunnel. In order to simulate the oscillating motion the airfoil is not moved with respect to the fluid, but the fluid oscillates while the airfoil remains fixed. The oscillating motion of the fluid is generated by acoustic waves, which are produced by two speakers in the side walls of the wind tunnel (Section 3.1). A NACA 0018 profile is used. The pressure difference over this airfoil is measured with two pressure transducers respectively on the top- and bottom sides. The stationary pressure difference corresponds to the time averaged signal. The oscillating difference is determined by means of a lock-in data analysis method, which uses the sinusoidal driving signal to the amplifier of the speakers as reference signal (Section 3.2).

Transversal acoustic velocity at the airfoil

Particular care is taken in determining the transversal acoustic velocity. When the frequency of the driving signal corresponds to the first transversal acoustic resonance of the test section, a complex acoustic field is found. This field is not merely determined by the first resonant transversal mode (Section 4.2). The transversal acoustic velocity at the location of the wing is calculated from the pressure difference at 2.5 times the half chord length upstream from the speakers. In an empty test section, the acoustic velocity agrees within 3% (uniform velocity of 41 m/sec) to the velocity at the wing position. It is assumed that the estimation of the acoustic velocity remains valid when an airfoil is placed in the test section (Section 4.3).


78

Dynamic response of NACA 0018

The dynamical behavior of the NACA 0018 airfoil is compared to the thin airfoil model. In this way the measurement method is verified. The thin airfoil model considers the airfoil as an infinitely thin two dimensional plate with a zero angle-of-attack. The dynamic motion is determined by a purely transversal oscillation, perpendicular to the main velocity (Section 2). The local stationary behavior is expressed in stationary pressure coefficient, which is the local pressure difference between the top- and bottom sides of the airfoil divided by the dynamic pressure in the wind tunnel (Section 5.1). The coefficient is compared to the thin airfoil model for several angles-of-attack. The differences can be explained by the finite thickness of the experimental airfoil (Section 5.2.1). The local oscillating behavior is expressed in oscillating pressure coefficient, where the local oscillating pressure difference between the top- and bottom sides of the airfoil is divided by the dynamic pressure in the wind tunnel and the ratio between the acoustic velocity and main velocity. The coefficient is expressed as a function of the Strouhal number, St , defined as the product of half chord length and angular frequency divided by the main wind velocity (Section 5.1). The oscillating pressure coefficients of the NACA 0018 for a zero angle-of-attack give consistently higher amplitude, 40%, compared to the thin airfoil model. The difference in phase increases for increasing Strouhal numbers. The deviations are also expected to be related to the finite thickness of the wing. Moreover, the oscillating pressure coefficient does not depend on the amplitude of the acoustic pressure perturbations. In addition, a small angle-of-attack does not influence the coefficient. However, if the angle-of-attack becomes large enough, a drop in oscillating pressure coefficient is observed for a high Strouhal number (Section 5.2.2).

Influence of a cavity in the wing

The impact of a cavity on the dynamical behavior is investigated. Firstly, the stationary behavior is not anti-symmetrical for the angle-of-attack anymore. The local stationary pressure coefficient decreases for negative angles-of-attack (Section 6.1.1). Secondly, the influence of the cavity on the oscillating pressure coefficient is considered at a zero angle-of-attack. An increment of the coefficient is found around Strouhal number St = 3.5. Moreover, for Strouhal numbers larger than St > 4.9, the slope of oscillating pressure coefficient as function of the Strouhal number increases. The phase agrees well to the phase of the original airfoil. To study the oscillating pressure coefficient in more detail, the influences of the amplitudes of the acoustic pressure waves and angles-of-attack are investigated as well. Decreasing the acoustic amplitude results to an increasing difference in phase for Strouhal number lower than ≤St 3.5. Next, the oscillating pressure coefficient is not influenced by a small positive or negative angle-of-attack. Nevertheless, if the angles-of-attack becomes larger a change in dynamical behavior is observed (Section 6.1.2).

General conclusion

Comparison between the local pressure differences on a NACA 0018 airfoil and the thin airfoil model shows that the measurement method is suitable for determination of the dynamical behavior of an airfoil. Difference between the theoretical and experimental values can probably be explained by the finite thickness of the airfoil. The method demonstrates a significant impact of the cavity on the dynamic response of the airfoil. Difference for both the stationary and oscillating behavior are observed.


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7.2 Recommendations

- The dynamical behavior of the NACA 0018 airfoil is compared to the thin airfoil theory. This model considers the airfoil as an infinitely thin plate, while the experimental airfoil has a finite thickness. It is suggested to consider a theoretical model which takes the thickness into account.

- The experiments in the wind tunnel are restricted to the first transversal

resonance frequency of the wind tunnel. Hence, only relative high Strouhal numbers can be obtained (3.0 < St < 11). It is suggested to develop another set of experiments, for instance were the airfoil moves with respect to the fluid, to study the dynamic response at a lower Strouhal number

- The pressure difference on the NACA 0018 airfoil is only obtained from the

two pressure transducers on the top- and bottom sides of the airfoil. Hence, only the local pressure differences are studied. It is recommended to investigate the difference between the local pressure difference and the integrated one.

- The acoustic velocity is an important parameter in this thesis. The acoustic field

in the wind tunnel is not fully understood. It is recommended to consider the acoustics by numerical calculation which includes wall vibrations and the finite length of the test section.

- In addition, probably the acoustic field in the wind tunnel consists of more

acoustical modes besides the first transversal mode. To investigate these modes more pressure transducers, placed anti-symmetrical around the cross-section of the wind tunnel, could be used.

- Not all data signals of the pressure transducers on the airfoil are investigated,

such as the pressure transducer at the leading edge and the second transducer in the cavity. To understand the dynamical behavior better, more time is ought to be spent on these signals.

- The dynamical behavior of the airfoil with cavity is determined for a scale

model. The results need to be scaled to obtain the dynamical behavior of the airfoil in reality. Nevertheless, this may give problems as the velocity has to be increased a large amount to preserve the Reynolds number. Consequently the

assumption of incompressible fluid is no longer satisfied ( 1)/( 22 <<= cUM ).

- More attention should be spent on the noise level of the signals. This gives an

indication of the linear relation between the response signal and the input signal.


80


81

Bibliography Chapter 1 [1] VortexCell2050, Fundamentals of actively controlled flows with trapped vortices,

Proposal No 12139, call identifier FP6-2003-Aero-1, 2003

Chapter 2 [2] F.C. Fung, An introduction to the theory of Aeroelasticity, Dover Publications, USA, Mineola,

2002, chapter 12 and 13 ISBN: 0-486-49505-1, Previously published: New York, Dover, 1969

[3] Pijush K. Kundu, Ira M. Cohen, Fluid Mechanics, third edition, Elsevier Academic Press, USA, California, 2004 ISBN-13: 987-0-12-178253-5 ISBN-10: 0-12-178253-0

[4] Th. Von Kármán, W.R. Sears, Airfoil Theory for Non-Uniform Motion, Journal of the Aeronautical Sciences, Vol 5, Number 10, August 1938

[5] M.A. Biot, Some Simplified Methods in Airfoil Theory, Journal of the Aeronautical sciences, Vol 9, No 5, March 1942

[6] Joseph Katz, Allen Plotkin, Low-Speed Aerodynamcis, second edition, Cambridge University Press, USA, New York, 2001 ISBN-13: 987-0-521-66219-2 ISBN-10: 0-521-66219-2

[7] T. Yao-tsu Wu, Swimming of a waving plate, California Institute of Technology, California, Pasadena, Report NO. 97-1, August 1960

[8] H.G. Küssner, I. Schwarz, The oscillating wing with aerodynamically balanced elevator, NACA Technical Memorandums, TM-991, October 1941 (Translated from Luftfahrtforschung, Vol 17, No 11-12, 1940)

[9] Watson Fulks, Complex Variables, An Introduction, Marcel Dekker Inc, USA, New York, 1993 ISBN: 0-8247-9079-0

[10] Churchill, Complex variables and applications, second edition, McGraw-Hill book company, Inc, USA, New York, 1960


82

Chapter 3 [11] Hilbert Transform, Specialized Transforms, Signal Processing Toolbox, The

MathWorks, Inc. 1994-2005

[12] M.L. Meade, Lock-in amplifiers: principles and application, Peregrinius, London, 1983 ISBN: 0-906048-94-X

[13] Maximiliano Osvaldo Sonnaillon, Fabián Jose Bonetto, A low-cost, high-performance, digital signal processor-based lock-in amplifier capable of measuring multiple frequency sweeps simultaneously, American Institute of Physics, review of scientific instruments 76, 024703-1, 2005

[14] Notes on Noise Reduction, Rice Unix Facility (RUF), PHYS 331: Junior Physics Laboratory I, 10 oct 1999

[15] Paul A. Temple, An introduction to phase-sensitive amplifiers: An inexpensive student instrument’, American Journal of Physics, Vol 43, No 9, September 1975

[16] (Afstudeerschrift) Maurice van de Nobelen, Detecteren van niet-lineariteiten in een systeem met behulp van de Hilbert transformatie, WFW 88.041, Technische Universiteit Eindhoven, Afdeling Werktuigbouwkunde, juni 1988

[17] Smith, J.O., Mathematics of the Discrete Fourier Transform (DFT), http://ccrma.stanford.edu/~jos/mdft/, 2007, ISBN 978-0-9745607-4-8.

Chapter 4 [18] A.P.Dowling, J.E. Ffowcs Williams, Sound and Sources of Sound, Ellis Horwood Limited,

West Sussex, England, 1983 ISBN 0-85312-527-9

[19] P.M.Morse, K.U. Ingard, Theoretical Acoustics, McGraw-Hill Book Company, USA, 1968

[20] A. Hirschberg, Elements of Acoustics, IACMA-International Advance Course on Musical Acoustics, Bologna, Italy, 2005

[21] Walter Eversman, The Effect of Mach Number on the Tuning of an Acoustic Lining in a Flow Duct, The Journal of the Acoustical Society of America, V48, No. 2, 1969

[22] Walter Eversman, Energy Flow Criteria for Acoustic Propagation in Ducts with Flow, The Journal of the Acoustical Society of America, V49, No. 6, 1970

[23] M.L. Munjal, Acoustics of Ducts and Mufflers, with application to exhaust and ventilation system design, John Wiley & Sons Inc, Canada, 1987 ISBN: 0-471-84738-0

[24] Robert A. Adams, Calculus – A Complete Course, fifth edition, Addison Wesley Longman, Toronto, Canada, 2003 ISBN: 0-201-79131-5

[25] Ir. W.F.J. Olsman, Post-graduate researcher TUe, private communication, January 2008


83

Chapter 5 [26] Ir. W.F.J. Olsman, Post-graduate researcher TUe, private communication, February 2008 Appendix A [27] S.E. Wright, The acoustic spectrum of axial flow machines, Journal of Sound and Vibrations,

Vol 45, no. 2, Mar 1976, pp 165-223

[28] Thomas F. Brooks, D. Stuart Pope, Michael A. Marcolini, Airfoil Self-Noise and Prediction, NASA Reference Publication 1218, 1989

Appendix B [29] Public Domain Aeronautical Software (PDAS) , Tables from Appendices I,II,III of

Theory of Airfoil Sections, Appendix I – Profiles, http://www.pdas.com/profiles.htm, 25 February 2008


84


85

Appendix A

Investigation of Airfoil self noise Under certain circumstances, like a particular velocity and angle-of-attack, a whistle sound is heard in the wind tunnel. This self noise results from coupling of flow instabilities and acoustic resonances of the wind tunnel. The phenomenon is investigated in this appendix. In the first section some background information is given, which explains how a whistle sound emerges. Next, experiments are performed, firstly on the NACA 0018 airfoil, subsequently on the airfoil with cavity.

A.1 Airfoil self noise

Self-noise from an airfoil arises due to interaction between the airfoil and small disturbances in de boundary layer. The occurrence of the whistle phenomenon can probably be explained by transition in the boundary layer from laminar to turbulent. As the Reynolds number,

µρ /cRe U= were c is the chord length, exceeds a critical value, small disturbances arise at a

certain point in the boundary layer on the airfoil. These disturbances are an initial phase of the laminar-turbulent transition and will develop to Tollmien-Schlichting waves which are converted by the flow towards the trailing edge. This is illustrated in figure 57a. At the trailing edge these waves will induce pressure disturbances which act as a dipole. The pressure waves are radiated outwards. As the frequency of the radiated pressure disturbances match to a resonance of the wind tunnel, the waves are amplified. The strong pressure waves reinforce the original disturbances in the boundary layer and an aero-acoustical feedback-loop is completed (see figure 57b). The waves are self-sustained and a whistle sound is observed [27][28].

(a)


86

(b)

Figure 57 (a) Airfoil self-noise. The wavelengths of the Tollmien-Schlichting waves are indicated by λ s,

the wavelength of the radiated whistle sound by λ acoustic. The speed of sound is represented by 0c and

the velocity of the main flow by U . (b) Feedback loop for airfoil whistling

A.2 Self noise of NACA 0018

Experiments are performed to investigate the whistle sound. Firstly, the whistle sound of the NACA 0018 is considered. The NACA 0018 is placed in the wind tunnel and a small angle-of-attack is given to the airfoil. The main velocity is subsequently varied between U = 6 m/sec and U = 21 m/sec. The responding whistle sound is measured by the pressure transducer at the top of the wind tunnel (location II, see figure 58).

Figure 58 Location pressure transducer II

The results of the airfoil self noise are presented in figure 59. In the left figures the strength of

the acoustic velocity divided by the main velocity, ) /(p' /' 0UcUu ρ= , is plotted as function of

the Strouhal number. This gives information about the strength of the whistle sound. In the figures at the right hand side the ratio between the width of the wind tunnel and the wavelength,

λ/2L , is plotted as function of the Strouhal number. This gives an indication of the shape of the acoustic modes. The upper two figures are the results obtained at an angle-of-attack of α = 1 degree, the middle two for α = 2.5 degrees, and the two at the bottom at α = 5 degrees. A certain critical velocity is required to obtain a whistle sound. When the velocity is lower than 6 m/sec no self-noise is noticed. By varying the main velocity a shift in frequency of the noise is noticed, these are given by different symbols in the figures. Each observed frequency has a peak at a certain Strouhal number.


87

10 15 20 25 30 35

2

4

6

8

10

St (= ω b / U)

|p| /

( ρ

c0 U

)angle-of-attack = 1

o

331 Hz

488 Hz

865 Hz

992 Hz

12 x10 -3

10 15 20 25 30 350

0.5

1

1.5

St (= ω b / U )

2L /

λ

angle-of-attack = 1 o

331 Hz

488 Hz

865 Hz

992 Hz

18 25 31 37

5

10

15

20

25

St (= ω b / U )

|p| /

( ρ

c0 U

)

angle-of-attack = 2.5 o

330 Hz

488 Hz

648 Hz

882 Hz

998 Hz

30 x10-3

10 15 20 25 30 350

0.5

1

1.5

St (= ω b / U )

2L /

λ

angle-of-attack = 2.5 o

330 Hz

488 Hz

998 Hz

648 Hz

882 Hz

15 18 210

1

2

3

4x 10

-3

St (= ω b / U )

|p| /

( ρ

c0 U

)


337 Hz

432 Hz

660 Hz

802 Hz

12 15 18 210

0.5

1

1.5

St (= ω b / U )

2L /

λ


337 Hz

432 Hz

660 Hz

802 Hz

Figure 59 AIRFOIL SELF-NOISE. The experiments are obtained at three different angle-of-attack, α = 10 (upper), α = 2.50 (middle), and α = 50 (bottom). In the figures to the left hand side the dimensionless pressure amplitude are plotted against the Strouhal number. The figures to the right represent the width of

the wind tunnel ( L2 ) divided by the wavelength ( λ ) as function of the Strouhal number.


88

At an angle-of-attack of 1 degree two frequencies are apparent, 331 Hz and 992 Hz. The first corresponds to the first transversal mode of the wind tunnel, the second to the third mode (figure 60). This can also be concluded from the right figure. At 331 Hz a half wavelength fits in the wind tunnel, λ/L ~ 0.5. At 992 Hz the ratio is ~1.5. This behavior corresponds to a dipole character of the pressure disturbances at the trailing edge. Dipoles originate from a force and require an acoustical velocity in order to generate acoustic energy. At the first and third mode the acoustic pressure has a node in the middle of the wind tunnel and the acoustic velocity an anti-node. As the trailing edge (source point) is located in the middle, the uneven modes are reinforced and a sound is observed. At the second resonance mode this is reversed, i.e. the pressure has an anti-node and the velocity a node. No sound is produced. Other measured frequencies at α = 1 degree are 488 Hz and 865 Hz. These peaks are not understood. At an angle-of-attack, α = 2.5 degrees and α = 5 degrees, the same behavior is noticed. Nevertheless, at α = 2.5 degree an additional frequency is noticed around 648 Hz. Furthermore, at α = 5 degrees the apparent peak around 993 Hz is diminished, and only a clear peak around 802 Hz is noticed.

Figure 60 First, second and third mode in wind tunnel section. For the first and third mode the acoustic pressure, 'p , has a node in the middle of the wind tunnel whereas the acoustic velocity, 'v , is has an anti-

node. In the second mode the acoustic pressure 'p has an anti-node in the middle and the acoustic

velocity, 'v , a node.


89

A.3 Self noise of airfoil with cavity

Subsequently, experiments on the airfoil with cavity are performed and the influence of this cavity on the whistle sound phenomenon is studied. Again, the arising noise is measured by the pressure transducer located at the top wall of the wind tunnel (location II, figure 61). The main velocity is varied between U = 6 m/sec and U = 21 m/sec.

.

Figure 61 location pressure transducer II The results are presented in figure 62. In the left figures the strength of the acoustic velocity

divided by the main velocity, ) /(p' /' 0UcUu ρ= , is plotted as function of the Strouhal

number. In the figures to the right hand side the ratio between the width of the wind tunnel and the wavelength, λ/2L , is plotted as function of the Strouhal number. The experiments are presented for three different angles-of-attack, α = + 1 degrees (upper figures), α = + 2.5 degrees (middle figures), and α = +5 degrees (lower figures). The experiments are also performed for negative angles-of-attack. However, at these angles no sound is observed in the velocity range 6 to 21 m/sec.


90

10 20 30 400

2.0

4.0

6.0

8.0

10.0

St (= ω b / U)

|p| /

(ρ c

U)

angle-of-attack = + 1o

335 Hz

860 Hz

992 Hz

12.0 x10-3

10 20 30 400

0.5

1

1.5

2L /

λ

St (= ω b / U)

angle-of-attack = + 1o

335 Hz

860 Hz

992 Hz

6 13 19 25 31 38 44 50

5

10

15

20

25

St (= ω b / U)

|p| /

(ρ c

U)

angle-of-attack = + 2.5 o

340 Hz

868 Hz

924 Hz

1737 Hz

30x10-3

6 13 19 25 31 38 44 500

0.5

1

1.5

2

2.5

3

St (= ω b / U)

2L /

λ

angle-of-attack = + 2.5 o

340 Hz

868 Hz

924 Hz

1737 Hz

13 25 38 50 63 75 88 101

50

100

150

200

250

300

St (= ω b / U)

|p| /

(ρ c

U)

angle-of-attack = + 5 o

337 Hz

864 Hz

992 Hz

2056 Hz

3810 Hz

350x10-3

13 25 38 50 63 75 88 1010

1

2

3

4

5

6

St (= ω b / U)

2L /

λ

angle-of-attack = + 5 o

337 Hz

864 Hz

992 Hz

2056 Hz

3810 Hz

Figure 62 AIRFOIL SELF-NOISE of the airfoil with CAVITY. The experiments are obtained at three different angles-of-attack, α = +10 (upper), α = +2.50 (middle), and α = +50 (bottom). In the figures to

the left hand side the dimensionless pressure amplitudes are plotted against the Strouhal number, St . The

figures to the right represent the width of the wind tunnel, L2 , divided by the wavelength, λ , as function of the Strouhal number.


91

The results of the NACA 0018 airfoil and the airfoil with cavity are compared. The Strouhal numbers (and frequencies) at which airfoil self noise occurs are for both airfoils roughly the same. However, apparent peaks may arise at different frequencies. The first peak around 340 Hz, which corresponds to the fist transversal mode in the wind tunnel, is observed for all the three angles-of-attack. In addition, it is seen that for the airfoil with cavity more higher frequencies arises, which were not noticed on the NACA 0018 airfoil.

A.4 Conclusion

The airfoil self noise is measured for three different angles-of-attack. At critical velocities and angles-of-attack a whistle sound is observed. Two clear peak values are found, which corresponds to the fist and third transversal acoustic modes of the wind tunnel. For an airfoil with cavity globally the same whistle frequencies as for the NACA 0018 are observed. This is logical if these frequencies are determined by the transversal acoustical resonances of the test section. In some cases however, the whistle frequencies are different (around 860 Hz instead of around 991 Hz). Furthermore, the whistle sound is only observed for positive angles-of-attack. For negative angles-of-attack no sound is noticed.


92


93

Appendix B

Mathematical techniques To solve the problem of the stationary and oscillating thin airfoil (Section 2.3 and 2.4), two mathematical techniques are used; acceleration potential and conformal mapping. The kinematics of the flow around the thin airfoil can be expressed by means of a potential flow, which is governed by the equation of Laplace. In this case, the real and imaginary part of an analytic complex function is describing the flow field. The conformal mapping method is a mathematical technique which is linked to this complex function theory. In this section first the acceleration potential is explained, thereafter the conformal mapping.

B.1 Acceleration potential

The overall idea is that in a frictionless flow the flow field can be described by introducing a complex acceleration potential. From this potential the acceleration can be calculated in each position in the flow field. The acceleration potential is proportional to the pressure, so once the acceleration potential is known the aerodynamic forces on an airfoil can be calculated. The motion of an inviscid flow is given by the Euler equation, which is in (B.1) adapted for a barotropic flow. In this flow the density is only a function of the pressure, i.e. )( pρρ = .

∫−∇=∇−=p

p

dpp

Dt

vD

0

1

ρρ

v

(B.1)

It is seen that the acceleration, DtvDarr

= , is a gradient of a scalar function. This function is

called the acceleration potential, φ .

φ∇≡a

r (B.2)

For an incompressible flow the velocity satisfies the continuity equation, 0=⋅∇ v

r. Hence, when

the divergence is taken on both sides of (B.1), it is seen that the acceleration potential φ satisfies

the Laplace equation.

02 =∇ φ (B.3)

In the case when a two dimensional flow is considered, 0// 22222 =∂∂+∂∂=∇ yx φφφ , and

the acceleration potential exists and is differentiable in the whole domain, this implies that a second function can be found which is orthogonal to this acceleration potential. This function is


94

called the acceleration streamfunction, ψ. Together with the acceleration potential this function forms the complex acceleration potential, w(z), which is an analytical function of the complex variable jyxz += ,

ψφ jzw +=)( in which jyxz += and 12 −=j 4) (B.4)

The relation between the acceleration potential, φ , and the streamfunction, ψ , can be shown

when the partial derivative with respect to the x- and y-coordinate is taken of complex potential.

x

jxdz

dw

x

z

dz

dw

x

w

constyconsty ∂∂

+∂∂

≡=

∂∂

=

∂∂

==

ψφ (B.5)

y

jydz

dw

yj

ydz

dwj

y

z

dz

dw

y

w

constxconstx∂

∂−

∂

∂≡⇒

∂

∂+

∂

∂≡=

∂

∂=

∂

∂

==

φψψφ

Comparing both functions and making a distinction between the real and imaginary component, the relation between the acceleration potential and streamfunction can be expressed as

yx ∂

∂=

∂∂ ψφ

, xy ∂

∂−=

∂∂ ψφ

(B.6)

Which are known as the Cauchy-Riemann equations. Using these equations it can be proven that also the streamfunction satisfies the Laplace

equation, i.e. 02 =∇ ψ . This is confirmed by

02

2

2

2

2

2

2

2

2

2

2

=∇=∂

∂+

∂

∂⇒

∂∂∂

−=∂

∂∂∂

∂=

∂

∂

ψψψ

φψ

φψ

yx

yxx

yxy (B.7)

Since the acceleration is related to the acceleration potential by xax ∂∂= /φ and ya y ∂∂= / φ ,

and to the streamfunction ψ by yax ∂∂= /ψ and xa y ∂−∂= /ψ , the derivative of the

complex acceleration potential, w(z), to the variable jyxz += can then be expressed as

yx jaadz

dw−= (B.8)

In conclusion, once the complex acceleration potential is known, the acceleration, a

r, of flow

field can be determined by differentiating the potential with respect to the complex coordinate, z .

4) Both i and j indicates an imaginary number, i.e. i

2 = -1 and j

2 = -1. However, both numbers are used

for a different application. While i indicates a phase shift in time for a harmonic oscillation, j is used to

separate the potential and its conjugate in the complex acceleration potential.


95

Often lines of constant ψ (streamlines) and constant φ (equipotential lines) are drawn. Since

the equipotential lines and streamlines are orthogonal these lines intersect each other at right angles everywhere. Moreover, streamlines are tangent to the vector of the acceleration, for this reason the streamlines are a useful tool to visualize the acceleration field (figure 63). Streamlines are governed by 0=ψd . This can also be written as

0=+−=∂∂

+∂∂

= dyadxadyy

dxx

d xy

ψψψ (B.9)

Hence,

=

x

y

a

a

dx

dy (B.10)

Figure 63 Vector of acceleration tangent to streamline ψ

The acceleration potential is directly related to the pressure. From (B.1) and (B.2), the acceleration potential can be described by

constdp

p

p

=+ ∫0

ρφ (B.11)

The constant term at the right hand side of the equation can be a function of time. However, since the boundary condition at infinity states that the parameters become steady, this term is a real constant.

Next, in linearized approximation the acceleration potential is considered as '0 φφφ += , the

density as '0 ρρρ += , and pressure as '0 ppp += where 0φ , 0ρ , and 0p are a reference

state of the undisturbed, uniform flow, and 'φ , 'ρ , and 'p are the fluctuations for which

1/' 0 <<ρρ and 1/' 0 <<pp . Substituting this in (B.11), where the reference state of the

acceleration potential is assumed to be zero, 0φ = 0, results to

( )

0...''

1'

'1

) (

'

0

2

000

'

00

0

0

0

=

+

+−+′=

++′+ ∫∫

+ ppp

p

dpdp

ρρ

ρρ

ρφ

ρρρφφ (B.12)

Likewise;


96

( )2

0

''

pOp

+−=′ρ

φ

Hence, the perturbations of the acceleration potential are indeed proportional to the pressure disturbance of the flow when higher order terms are neglected. In fluid dynamics often a velocity potential is used instead of an acceleration potential. In that

case, the flow field is described by the velocity potential, φ~

, and the stream function ψ~ .

Likewise, a complex velocity potential can be defined as ψφ ~~~ jw +≡ .

An acceleration potential is used instead of a velocity potential since the first one is continuous everywhere while the latter one has a discontinuity in the wake of the airfoil. The wake behind an airfoil is represented as an infinitely thin sheet on which vorticity is concentrated. These vortices cause a discontinuity across the sheet in tangential velocity (see figure 64). The velocity potential is discontinuous across this surface, whereas the pressure and thus the acceleration potential are continuous.

Figure 64 Airfoil where the wake is represented by a vortex sheet. Along this sheet the velocity is discontinuously.

B.2 Conformal mapping

To work out the dynamics of a thin airfoil in z-plane, jyxz += , is difficult. For this reason, a

conformal mapping method is used to transform the problem, together with the boundary conditions, into a problem for which the solutions are already known. This method is based on complex functions and is a mathematical tool to simplify problems such as the Laplace equation. A conformal mapping transforms a point in the complex z-plane, jyxz += , to a point in the

complex ζ-plane with ηξζ j+= (see figure 65). Firstly, this technique is explained, next a

suitable conformal mapping for the problem of an oscillating airfoil is proposed.


97

Figure 65 Conformal mapping method. Point α and curve C through α in z-plane is transformed to

point β and curve Γ through β in ζ-plane.

Consider a function )(zF in region R in the z-plane which is analytic, thus differentiable in

R. A point α is located in this region and a curve C can be drawn which is also located in R and passes through this point. This curve is given by function )(tg .

)( : tgzC = and )( : 1tgz =α (B.13)

A function ηξζ izF +== )( is considered which maps point α to a corresponding point β

in the ζ-plane. Obviously, the line C is mapped as well to a line Γ in the ζ-plane.

) )( ( : tgF=Γ ζ and ) )( ( : 1tgF=ζβ (B.14)

The transformation is called conformal mapping and corresponding points or lines are named images of each other. It is assumed that this conformal mapping function has an inverse,

)(1 ζ−= Fz .

To examine the relation between the lines C and Γ near t = 1t , a vector tangent to line C at

point α is considered. This vector is described as

11 tttt t

g

t

C

==

∂

∂=

∂

∂ (B.15)

The image of this vector in point β is then

( ) ( )111

t

g )(

tttttt t

FtgF

tt ===

∂

∂

∂

∂=

∂

∂=

∂

Γ∂ α (B.16)

Hence, the vector tangent to line Γ is taF ∂∂ /)( times the vector tangent to line C at point α

in the z-plane. Note that this factor does not depend on the choice of line C but depends on the position α , so all vectors at point α are multiplied by this factor when transformed to the

ζ-plane.

In addition, if (B.16) is approached as ( ) ztF ∆∂∂≈∆ /)(αζ , two line elements at point α are

multiplied by the same factor when transformed, that is to say both lengths of the segments are

multiplied by tF ∂∂ /)(α and both are rotated by ( )tF ∂∂= /)(arg αφ . Consequently, the angle


98

between the two line elements at point α in the z-plane remains equal to the angle between the

images in the ζ-plane. This is an important property of the conformal mapping technique. In a point the geometrical properties are preserved. In spite of this, a large figure can loose its original shape as the factor tF ∂∂ /)(α depends on point α and will change for a different point in the

z-plane. A common conformal mapping in aerodynamic studies is the Joukowski transformation in

which a straight line segment [ ]1,1−∈x in the z-plane, jyxz += , is mapped into a circle

with radius one in the ζ-plane, ηξζ j+= . The function is:

( )

+== −

ζζζ

1

2

1z

1F (B.17)

This conformal transformation is used in the further calculations. A point located on the unit

circle ( θζ je= ) corresponds to the point )cos(θ=x on the line segment where

)/arctan( ξηθ = . In each point on the circle the angle between the two adjacent line elements is

preserved except at two points; the leading edge and trailing edge. These two points contain a

singularity. The space outside the circle in the ζ-plane is mapped into the entire z-plane. (See figure 66)

Figure 66 Joukowski Transformation


99

Appendix C

Boundary conditions thin airfoil model adapted to complex ζ-plane The acceleration potential has to satisfy boundary conditions. When the acceleration potential is

mapped into the conformal ζ-plane the boundary conditions have to be transformed to the

conformal ζ-plane as well. (Section 2.2) For this reason the boundary conditions for both the stationary and oscillating airfoil are adapted. I. The first condition states that the velocity is finite at the trailing edge (Kutta-Joukowski

condition). An equivalent form of this condition is that no pressure discontinuity exists at the trailing edge. An acceleration potential should be chosen which is continuous and remains finite at this point.

)0,1( == θφ r is finite (C.1)

IIa. The boundary condition for the fluid velocity normal to the airfoil remains the same.

Hence,

nvnv fluidairfoil

vvvv⋅=⋅ (C.2)

Next, the additional boundary conditions for the oscillating airfoil are considered. IIb. The oscillating velocity boundary condition at the airfoil surface is not automatically

satisfied by introducing an acceleration potential. The velocity normal to the airfoil can be related to the y-component of the acceleration. For this, the complex representation for harmonic oscillations for the acceleration potential and velocities is used.

tie

ωφφ = tieuu

ω ˆ'= tievv

ω ˆ'= (C.3)

Then, the vertical acceleration can be expressed as

x

vU

t

v

ya y ∂

∂+

∂∂

=∂∂

=''

'φ

(C.4)


100

Or for harmonic perturbations

tititie

x

vUevie

ωωω ωφ

∂∂

+=∂∂ ˆ

ˆ y

ˆ (C.5)

This first order linear difference equation is solved for tievv

ωˆ'= . The homogeneous solution for this equation results to:

( )Uxi

h

tih

tih

Cev

eviex

vU

ˆ

0ˆˆ

ω

ωω ω

−=⇒

=+∂

∂

(C.6)

Subsequently, the non-homogeneous equation is solved by variation of constant C .

Here, C is assumed to be a function of x , that is to say )(xCC = .

( ) ( )

( ) ( )

( )U x

U xU x

U xU x

ˆ

ˆ

ˆ

ω

ωω

ωω

ω

ω

ω

i

ii

ii

ex

CUvi

x

vU

ex

CUCei

x

vU

CeU

ie

x

C

x

v

−

−−

−−

∂∂

=+∂∂

⇒

∂∂

=+∂∂

⇒

−∂∂

=∂∂

(C.7)

Substitution this in (C.5) gives

( )

( )

( ) εφ

φ

φ

ωε

ω

ω

∫∞−

−

∂∂

=⇒

∂∂

=∂∂

⇒

∂∂

=∂∂

x

Ui

i

i

deyU

eyUx

C

ex

CU

y

/

U x

U x

ˆ1 C

ˆ1

ˆ

(C.8)

Here ε substitutes the x parameter in the integral. Hence, the vertical velocity

component is given in (C.9).

( )

( ) ( )

ˆ1

ˆ

-

/

Uxi

x

Ui

Uxi

edeyU

Cev

ωεω

ω

εφ −

∞

−

∂

∂=

=

∫ (C.9)

The boundary condition states that on the surface of the airfoil this velocity must be

equal to

titieyi

x

YU

t

Yevv

ωω ω ˆ ˆ' =∂∂

+∂∂

== (C.10)


101

for the specific situation where the motion of the plate is a pure vertical translation. III. The third boundary condition prescribed the vertical acceleration of the flow on the

airfoil, which is directly imposed by the complex acceleration potential,

yx jaadzdw −=/ . However, transformation of the complex acceleration potential

from the z-plane to the ζ-plane will alternate this condition. The derivative of the

complex acceleration function, tieww

ω ˆ = , can be written as

dz

d

d

dw

dz

dw ζζ

= or ζ

ζζζ d

dF

dz

dw

d

dz

dz

dw

d

dw )(1−

== (C.11)

Where )(1 ζ−F is the Joukowski transformation function

The acceleration of the fluid on the thin airfoil is equal to ( ) )0,(/' xy dzdwa = , likewise

the corresponding ‘acceleration’ in the ζ-plane is given by ( ) θζζ jen ddwa == /' 5). Thus,

the second part of (C.11) can be rewritten as

ζ

ζd

dFaa yn

)(''

1−

= (C.12)

The acceleration, ya' , is multiplied by the scale factor ζddF /1− . This factor is equal

to θsin , see (C.13)

θζζ

θ

ζζ θθ

sin12

111

2

1 2

2=−=−= −

==

j

ee

ed

dz

jj

(C.13)

Where ( )θθθθ

θ jjjj

eejj

ee 212

1

2)sin( −

−

−=−

=

The acceleration and image acceleration are described in complex notation

ti

yy eaaω

ˆ' = θζ

ω

ζ je

tinn

d

dweaa

=

== ˆ' (C.14)

The amplitude of the image acceleration normal to the airfoil in the ζ-plane can

therefore be determined by multiplying the amplitude of the acceleration in the z-plane

by the factor sin θ, of which the absolute signs are omitted to take the sign of the normal on the circle into account.

θθ sin),0,(ˆ),cos,1(ˆ 1txatxra yn === − (C.15)

5) Actually, it is not right to call the image a’n of the acceleration a’y also an acceleration vector since it is transformed into another plane. However, it is chosen to use this name and same symbol for acceleration to demonstrate that these terms are corresponding to each other.


102

IV. Finally, the last boundary condition demands that the perturbations in the acceleration

tend to zero at infinity. This condition is satisfied when the acceleration potential becomes a constant.

( )( ) constyxyx

=∞→or

,lim φ (C.16)

Considering the ζ-plane this condition is unchanged, i.e. ( )( ) const=∞→ηξηξφ

or

,lim


103

Appendix D

Determination constants A and B oscillating thin airfoil The flow around an oscillating thin airfoil is described by a complex acceleration potential. This

potential, tieww

ω ˆ= , consists of two pressure dipoles. In the complex ζ-plane, one dipole is located at the leading edge and is described by 1/ +ζjA , the other is located at the origin of the

unit circle and is given by ζ/jB . Here A and B are constants which needs to be determined

by boundary conditions. This section determines the constants A and B in detail. First the constant B is solved, hereafter constant A .

D.1 Determination constant B

The constant B is obtained from the normal component of the acceleration in the ζ-plane, ti

nn eaaω ˆ' = , on the unit circle i.e. r = 1 and )cos(2 11 θrr = .

( )θφφ

ζ

sinˆˆ

ˆ

1

2

1

Brr

a

r

n −=

∂

∂=

∂∂

===

(D.1)

Since the first term of the complex acceleration potential, )1/( +ζjA , does not impose an

acceleration normal to the circle this term is omitted. Boundary condition III (Appendix C) states

( )θsinˆ0

22yStUan −= (D.2)

From this it is seen that the constant B should be equal to

022yStUB = (D.3)

D.2 Determination constant A

The determination of the constant A involves more calculation.


104

The constant A is obtained in the z-plane. The value can be determined by the kinematic condition for the velocity, which is given earlier in (C.9) (Appendix C). In the integral the x parameter is substituted byε .

( ) ( ) εφ εωω

dey

eU

v

x

UiUxi

∫∞

−

∂

∂=

-

/ˆ1

ˆ (D.4)

In this equation the vertical velocity is related by y∂∂φ . Using the Cauchy-Riemann equation

(Appendix B) this term can be expressed in a streamfunction, x∂∂ψ . Moreover, the relation

UUbSt // ωω == is used, and a point on the airfoil is set as x = -1 + λ for 0 ≤ λ ≤ 2. Equation (D.4) then, can be rewritten as

( )

( )ε

εψ

εψ

εεψ

λε

λ

λελ

deU

e

deeU

v

iStiSt

iStiSt

∫

∫+−

∞

+−−

+−

∞

+−−

∂

∂+

∂

∂−=

∂

∂−=

1

-

211

1

-

1

ˆˆ

ˆ1ˆ

(D.5)

A infinitely small value can be given to the parameter λ . This equation is solved by integrating

in parts around the leading edge as this point contains a singularity where x∂∂ 1ψ tends to

infinity. This will result to (D.6) in which the limit λ → 0 is set.

[ ]

∫

∫∫−

∞−

−

∞−

−

∞−

−

∞−

∂

∂−+−=

∂

∂

−+

−−−=

1

21

1

1

2

1

1

1

1

ˆˆ

ˆ

ˆ ˆˆˆ

εε

ψψ

ψ

εε

ψεψψ

ε

εεε

diSteU

e

U

deU

edeiSt

U

ee

U

ev

iStiSt

iStiSt

iStiSt

iStiSt

(D.6)

The vertical velocity is subsequently solved in steps. First, the “ U/ˆ1ψ− ”-part is determined.

Subsequently the integration is solved. Note that the streamfunctions are written in the ζ-plane and have first to be transformed to the z-plane.

The first part of the velocity contains the acceleration streamfunction, 1ψ , which was given in

the ζ-plane as ( ) 111 /cosˆ rA θψ = . At the unit circle, r = 1 and )cos(2 11 θrr = , this function is

constant, which was demonstrated earlier in section 2.2 (equation (2.16)).

2

ˆ1

A=ψ (D.7)

Transforming to the z-plane, this term remains the same as it is independent of the polar coordinates. Next, the integral of the right hand side of (D.6) is considered. The acceleration streamfunctions

1ψ and 2ψ are transformed back to the z-plane. For this the inverse Joukowski transformation

is needed.


105

1)( 1

2

1)(

21 −±==⇔

+== −

zzzFFz ζζ

ζζ (D.8)

To consider the flow around the singularity point, the acceleration streamfunctions are

investigated on the negative ζ-axis, accordingly 12 −−= zzζ . The parameters r , 1r , θ and

1θ become ζ−=r , 11 −−= ζr , and θ = 1θ = π. Hence, the acceleration streamfunctions are

equal to

( ) ( )

ζζ

θψ

θψ

−−

=−−

−=

==

BA

r

B

r

A

1

cosˆ

cosˆ

2

1

11

(D.9)

Transforming this the z-plane results to

)1zB(z

)1(

ˆ 1)1(

ˆ

2

22

21

−+=

−−=

+−−=

zz

B

zz

Aψψ

(D.10)

After the functions are transformed to the z-plane, they are substituted into the integral of (D.6)

and the integral is worked out. Again, first for the part containing the 1ψ -term and later for part

containing the x∂∂ 2ψ -term.

( )

∫

∫

∫

∫∫

∞−

∞−

∞−

−

∞−

−

∞−

−+

−=

−−−−

=

−−+−=

−−+=

1

1

2

12

1

2

1

1

1

11

2

12

)11(

11

11

ˆ

εεε

εε

εε

εεε

εεε

εψ

ε

ε

ε

εε

deA

dA

e

dA

e

dA

ede

iSt

iSt

iSt

iStiSt

(D.11)

This integral is subsequently solved using the zero order and first order of the modified Bessel

functions of the second kind, )(0 izK and )(1 izK . The definitions of these modified Bessel

functions are given in (D.12) and (D.13).

∫∞ −

−=

12

0

1)( ε

ε

ε

de

izKiz

(D.12)


106

∫∫∞

−∞ −

−−

−+

=−

=

−=

12

12

01

1

1

1

1

1

)(

)()(

εεε

εε

ε

ε εε

dede

izd

izdKizK

iziz

(D.13)

Hence,

( )

( ) ( )

−+−=

−+−

−−=

+−+

−=

−+

−=

−

∞−

∞−

∞ ∞−−

∞−

−

∞−

∫

∫ ∫

∫∫

iSt

iStiSt

iStiSt

iStiSt

eiSt

iStKiStKA

eiSt

deiStKA

dedeA

deA

de

1

2

1

1

1

2

1

1

2

1

11

2ˆ

01

112

1

1 1

1

1

1

εε

εε

εε

εε

εεεε

εεε

εψ

(D.14)

Likewise, the solution of the integral containing the x∂∂ 2ψ -term can be expressed in terms of

the modified Bessel functions of the second kind.

( )

( )

+−−=

−

−=

−−=

−+=

∂

∂

−

∞−

∞−

−

∞−

−

∞−

∫

∫∫

iStKeiSt

B

iStKeiSt

B

deB

dBede

iSt

iSt

iSt

iStiSt

1

1

1

12

1

2

1

2

1

1

11

11

ˆ

ε

ε

εα

εε

ε

εε

εε

εψ

(D.15)

Finally, substituting the results in (D.6) the vertical velocity can be rewritten as

−+

−+−+

−=

∂

∂−+−=

−−

−

∞−

−∫

iStiStiSt

iStiSt

eiSt

iStKBeiSt

iStKiStKA

iStU

e

U

A

diSteU

e

Uv

1)(

1)()(

22

ˆˆ

ˆˆ

101

1

21

1 εε

ψψ

ψ ε

(D.16) This is simplified to

−+

+−=

iStUiStKe

UBiStKiStKe

U

iStAv

iStiSt 1)(

1)()(

2ˆ

101 (D.17)


107

According to boundary condition II (Appendix C), the vertical velocity of the fluid on the airfoil

is equal to 0 iUky . After also the constant B is filled in, the constant A can be determined.

( )

)( 2

)()(

)(2

)()(2

1)(

1ˆ

02

01

1022

01

1

StCyStiU

iStKiStK

iStK

iSt

yStU

iStKiStKeU

iSt

iStUiStKe

UBv

AiSt

iSt

−=

+=

+−

−−=

(D.18)

Here the function )()(/)()( 011 iStKiStKiStKStC += is called the Theodorsen’s function. [2]


108


109

Appendix E

Geometry NACA 0018

0 20 40 60 80 100-50

-40

-30

-20

-10

0

10

20

30

40

50

x

y

NACA 0018

Figure 67 Geometry NACA 0018 [29]

NACA 0018

Stations and ordinates given in per cent of airfoil (chord) x y dy/dx

0.0000 0.0000 ******** 0.5000 1.8320 1.7729 0.7500 2.2273 1.4246 1.2500 2.8409 1.0738 2.5000 3.9221 0.7162 5.0000 5.3320 0.4543 7.5000 6.2999 0.3312

10.0000 7.0242 0.2531 15.0000 8.0177 0.1527 20.0000 8.6063 0.0866 25.0000 8.9119 0.0379 30.0000 9.0026 -0.0001 35.0000 8.9229 -0.0307 40.0000 8.7045 -0.0559

x y dy/dx

45.0000 8.3711 -0.0769 50.0000 7.9410 -0.0947 55.0000 7.4286 -0.1099 60.0000 6.8451 -0.1232 65.0000 6.1987 -0.1351 70.0000 5.4959 -0.1459 75.0000 4.7405 -0.1562 80.0000 3.9347 -0.1661 85.0000 3.0788 -0.1762 90.0000 2.1716 -0.1868 95.0000 1.2098 -0.1981

100.0000 0.1890 -0.2105 L.E. radius = 3.570 percent chord


110


111

Appendix F

Photos experimental setup

Figure 68 Separated test section of the wind tunnel. The speakers are placed on the left and right side walls, the box over the speaker to the right is removed. In the middle an airfoil with cavity is placed. A fixed angle-of-attack can be selected by rotation of the airfoil.


112

Figure 69 Details of the experimental wings. To the left, NACA 0018 airfoil with Kulite semiconductor pressure transducers. Next figure, airfoil with a cavity. Inside the cavity two pressure transducers are placed. Figures to the right from top to bottom, details of the Kulite semiconductor pressure transducers (diameter transducers is 2 mm), cross section of NACA 0018, and detail of the cavity which geometry is marked by the white line.


113

Appendix G

Lock-in method To extract information at one specific frequency out of a noisy signal the lock-in method is a useful technique. The method is based upon the orthogonality of sinusoidal functions. An integral over the product of two sinusoidal functions with equal frequency will result into a value, while other multiplied signals with different frequency, like noise, will vanish when they are integrated. Optimal results are obtained when the integral is taken over an integer number of time periods. A practical application which is related to this method is the lock-in amplifier, which extracts information superposed on a carrier wave out of a noisy environment. The signal is multiplied to a reference signal which is tuned to the same frequency as the carrier wave. In this way the information on this wave is filtered out [12]. The lock-in method is used to analyze the data signals from the pressure transducers on the wing and on the size walls of the wind tunnel (Section 3.2). This section considers first the mathematical principles of the Lock-in method. A Hilbert transformation appears to be required, for this reason this transformation will be explained in the second section.

G.1 Mathematical principles

The basic principles of the Lock-in method are given in a block diagram in figure 70. The response signal of the experimental system is given by )(tf and the reference signal by )(tr .

This is the driving signal to the amplifier of the speakers. The lock-in method multiplies and integrates both signals. The output signal, )(tg , contains information about the amplitude of

the response signal at the reference frequency and the phase shift between the response signal and reference signal.

Figure 70 Block diagram of Lock-in method implemented in an experiment

It is assume that the reference signal )(tr is a sinusoidal function with an amplitude 1=refv ,

phase refφ , and frequency refω .


114

( )refref ttr φω += sin)( (G.1)

The response signal from the experimental system is given by )(tf . The amplitude v , and

phase, φ , are depending on the frequency, ω .

( ))(sin)()( ωφωω += tvtf (G.2)

Using goniometry it is seen that a new signal with two components will result when the sinusoidal functions are multiplied, one component with the difference between the two frequencies and one with the summation of those.

( ) ( )( ) ( )[ ]))(()(cos))(()(cos)(

2

1

sin)(sin)()()(

refrefrefref

refref

ttv

ttvtrtf

φωφωωφωφωωω

φωωφωω

+++−−+−=

++=

(G.3) The output signal, )(tg , can be decomposed in a Fourier expansion of sinusoidal functions.

∑∞

=

++=1

0 )]sin()cos([2

1)(

n

nnnn tbtaatg ωω (G.4)

in which for any integer number n and m

02

1a ; mean average of )(tg

∫=mT

nn dtttfmT

a

0

)cos()(2

ω ; even Fourier coefficients of )(tg (G.5)

∫=mT

nn dtttfmT

b

0

)sin()(2

ω ; odd Fourier coefficients of )(tg (G.6)

The time period for one oscillation is given by T, and the angular frequency is equal

to ( )Tnn πω 2 = .

The averaged value of )(tg will be removed, i.e. 0021 =a . Besides, the desired signal is a

sinusoidal function at one specific frequency. For this reason, the lock-in method satisfies to consider only the first order of the Fourier series, i.e. 1=n .

The lock-in method calculates the coefficients, 1a and 1b , by replacing the cos-term in the

integral of (G.5) by the reference signal, ( )refrefref tvtr φω += sin)( , or in the case for the odd

coefficient by 90 degrees phase shifted reference signal, ( )refrefref tvtr φω +−= cos)(* . This ½

π phase shifted signal is represented by “ * ”. The output signal, )(tg , becomes

)(*)()( 11 trbtratg += (G.7)

In which


115

∫=mT

dttrtfmT

a

0

1 )()(2

; even coefficients of )(tg (G.8)

∫=mT

dttrtfmT

b

0

1 )(*)(2

; odd coefficients of )(tg (G.9)

In the integration of (G.8) and (G.9) the components of the response signal, )(tf , at the

reference frequency remain. That is to say, the amplitude and phase of the response signal

converts to fref vv =)(ω and fref φωφ =)( . The coefficients 1a and 1b are a function of this

amplitude and phase.

The output, )(tg , can be rewritten into one periodic function with frequency refω . Euler’s

equation yields,

[ ][ ][ ])(

11

11

11

)(Re

)cos()sin(Re

)(*)(Re)(g

refti

refrefrefref

eiab

tbta

trbtrat

φω

φωφω+

+−=

+−+=

+=

(G.10)

Hence, the amplitude of the output signal is

fvbaA =+= 21

21 (G.11)

and the phase difference

πφφπφ +−=+

= reff

b

a

1

1arctan (G.12)

In this mathematical analysis a 90 degrees phase shifted reference signal is needed to calculate

the odd coefficient 1b . To create this 90 degrees phase shift a Hilbert transform is applied. This

transformation will be explained in the next section.

G.2 Hilbert transform

A Hilbert transform can be used to generate two signals out one reference signal. A property is that both new signals contain the same information as the reference signal and are 90 degrees phase shifted with each other. The definition of a Hilbert transform, H is given as

∫∞

∞−−

=∗== ττ

τππ

dt

rtr

ttRtr

)(1)(

1)()]([Η (G.13)

In which the capital letter, )(tR , denotes the Hilbert transformed. In time domain, a Hilbert

transformation is the convolution operation, ‘∗’, between the signals )(tr and tth /1)( π= . A


116

convolution is a mathematical operator which calculates the overlap between one function and the reversed, translated second function. The Fourier transform of the Hilbert function, tth /1)( π= , contains a sign function

)sgn()]([)( ωω ithH −==F (G.14)

Here, F [..] denotes the Fourier operator, and sgn represents the sign function

<−

=

>

=

0for ,1

0for ,0

0for ,1

)sgn(

ω

ωω

ω (G.15)

For positive frequencies the Hilbert transformed function is multiplied by – i and a phase shift of –½ π is given, for negative frequencies the function is multiplied by + i which result in a

phase shift of + ½ π. Here i represents the complex number, i2 = -1. The amplitude and

phase shift of the original signal are maintained, but an extra phase shift of a quarter-cycle is added to the transformed signals. The function Hilbert in Matlab works on this principle. It is used to change a signal, )(tr , to a

causal function, )(ts , which has some benefits in signal processing. A signal is called causal if it

satisfies the condition )(ts = 0 for t < 0. The real part of the filtered signal is still the original

function, in contrast to the imaginary part, which is the original function with a phase shift of 90 degrees.

To illustrate this, an original signal given by tietr

ω=)( is assumed.

<

>=

− 0for t ,e

0for t ,)(

ti

tie

trω

ω

(G.16)

The Hilbert transformed is multiplied by ± i , depending on the sign of the time.

<

>−=

− 0for t ,ie

0for t ,)]([

ti

tiie

trω

ω

H (G.17)

The filtered signal, )(ts , is formed by adding the original signal with the multiplication of the

Hilbert transformed signal and the complex number i.

)]([ )()( tritrts H+= (G.18)

<=+

>=−+=

−− 0for t 0)(

0for t 2)()(

titi

tititi

ieie

eieiets

ωω

ωωω

Hence, the filtered signal, )(ts , is causal and its imaginary part, πω 21 )]([ ±= tietrH , and real

part, )( tietr

ω= , are equal in strength but 90 degrees shifted in phase.


117

Appendix H

In-approximation piezo pressure transducer PCB The in-approximation of a piezo pressure transducers (PCB 1510) located in the wall of the wind tunnel is investigated. Figure 71 represents the output signal, )(tg , of the lock-in method

for this pressure transducer. The amplitude in [V] is plotted as a function of the measuring time in [sec]. In one situation the integration is taken over an integer number of oscillating periods (solid line), in the second situation the integration is taken over an arbitrarily chosen time interval (dotted line). The relative error for the piezo transducer PCB after 1 sec is 0.15%.

10-3

10-2

10-1

100

101

102

103

0.33

0.34

0.35

0.36

0.37

0.38

0.39

Measuring time [sec]

Am

plit

ude [V

]

PCB 1510

with integer periods

arbitrarily time interval

0.343

Figure 71 Investigation of the inexactness lock-in method of a piezo pressure transducer in the wall of the wind tunnel (PCB 1510). The measuring time in [sec] is given on the x-axis in log scale, the amplitude in [mV] on the y-axis. The solid line is the output signal for the lock-in method taken on an integer number of periods, the dotted line is the output signal with a random time interval. The main velocity in the wind tunnel is 41 m/sec.


118


119

Appendix I

Higher order acoustical modes in wind tunnel The acoustic field in the wind tunnel section involves higher order modes. These modes are considered in the calculation of the acoustic field (section 4.1.2). This appendix considers the determination of these modes. The wind tunnel section is considered which has a square cross section with width L2 . The x-axis is parallel to the midline of the wind tunnel, and the yz-plane corresponds to a plane normal

to the x-axis, so y = ± L and z = ± L corresponds to the walls. See figure 72. The section is

assumed infinite long, hence no boundary condition in the x-direction are applied.

Figure 72 Geometry wind tunnel section. Two speakers are placed on the side walls at x = 0. A harmonic wave in the wind tunnel can be described by considering a separable function of the following form

[ ]tiezhygxfp

ω)()()(Re'= (I.1)

This separable function is substituted in the wave equation, 0')/')(/1( 2222

0 =∇−∂∂ ptpc , this

yields


120

0)(

)(

1)(

)(

1)(

)(

1

0'''

'

2

2

2

2

2

2

2

0

2

2

2

2

2

2

2

2

0

2

=∂

∂+

∂

∂+

∂

∂+⇒

=∂

∂+

∂

∂+

∂

∂+

z

zh

zhy

yg

ygx

xf

xfc

z

p

y

p

x

pp

c

ω

ω

(I.2)

The function )(yg is a function of y only, it is independent of x and z ,

2

2

2 )(

)(

1yk

y

yg

yg−=

∂

∂ (I.3)

in which 2yk is a constant. A possible solution for )(yg is of the form,

yik yeyg =)( . The

Boundary condition states that the normal velocity is zero at the wall. Using the momentum

equation in the y-direction, 0/')/1(/' 0 =∂∂+∂∂ yptv ρ , this boundary condition is equivalent to

0)('

=

∂∂

=

∂∂

±=±= LyLyy

yg

y

p (I.4)

From which results that the function )(yg is

( )2

)1(

sin)(−+= m

mym ykygπ

(I.5)

Where L

mk my

2

π= and m a positive integer, m = 0, 1, 2, ... .

The same procedure can be done for the function )(zh . This function is

)2/)1(sin()( −+= nzkzh nyn π , where Lnk nz 2/ π= for positive integer n . Hence, (I.2)

becomes

0)()()2(

)(

0)(

)(

1

22

2

2202

2

2

2

2

0

2

2

2

=

+−+

∂

∂⇒

=

−−+

∂

∂

xfnmL

kx

xf

kkcx

xf

xfnzmy

π

ω

(I.6)

Where the wavenumber 00 / ck ω= . The solution in complex notation is given by

xik

mnx

xik

mnxmnmnxmnx eAeAxf

)(+−−+ += (I.7)

In which its wavenumber is equal to ( )22

2

220

)2(nm

Lkk mnx +−=

π


121

Here the amplitudes ±mnA are determined by the boundary conditions. Accordingly, an acoustic

wave in the wind tunnel has the form of a mode

( ) ( )( )[ ]

[ ] Re

zsin y sinRe'

)(

)(

2

1)-(n 2

1)-(m

xkti

mnx

xkti

mnx

tixik

mnx

xik

mnxnzmymn

mnxmnx

mnxmnx

epep

eeAeAkkp

+−−+

−−+

+=

+++=

ωω

ωππ

(I.8)

The amplitude of the mn -th mode traveling in positive x-direction is represented by

+−−+ ++= mnx

n

nz

m

mymnx Aykykp 2

)1(

2

)1(

)sin()sin(ππ

, and for the negative x-direction by −−−− ++= mnx

n

nz

m

mymnx Aykykp 2

)1(

2

)1(

)sin()sin(ππ

. The overall solution for the wind tunnel is a superposition of the different modes described by this equation,

∑∑

∞

=

∞

=

=0 0

''n m

mnpp

(I.9)


122


123

Appendix J

Data signals The Lock-in method subtracts information at a desired frequency out of the response signal. Noise is filtered out. However, the rough, unmodified data signal gives information about the smoothness of the signal and gives an indication to which level the signal is affected by noise. This appendix gives the data signals of the pressure transducers on the top side of the airfoil. First the data signal of the NACA 0018 is considered, thereafter the signal of the airfoil with cavity. In addition, the data signal in the cavity is discussed.

J.1 Data signal NACA 0018

Details of the data signals of the pressure transducer at the top of the NACA 0018 are presented in figure 73 for three different Strouhal numbers, St = 3.5, St = 4.9 and St = 11.0. The response signal, )(tf , and the output signal, )(tg , of the lock-in method are given.


124

Figure 73 Details of the data signals of pressure transducer on top of the NACA 0018. The response

signal, )(tf , and the output signal of the lock-in method, )(tg are given for three different Strouhal

numbers, St = 3.5, St = 4.9, and St = 11.0. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 126 Pa

The response signal, )(tf , is undisturbed for Strouhal numbers from St = 4.9 to St = 11.0.

Increasing the velocity ( St < 4.9), more fluctuations are measured and the response signal is not a pure sinus anymore.


125

J.2 Data signal airfoil with cavity

Next, the data signal of the airfoil with cavity is considered for different Strouhal numbers. Again the response signal of the pressure at the top side is taken. The data signal is presented in figure 74 for Strouhal numbers St = 3.5, St = 4.9 and St = 11.0. The response signal, )(tf ,

and the output signal of the lock-in method, )(tg , are given.

Figure 74 Details of the data signals of pressure transducer on top of the wing of the AIRFOIL WITH

CAVITY. The response signal, )(tf , and the output signal of the lock-in method, )(tg are given for

three different Strouhal numbers. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic perturbation is | p ’| ~ 126 Pa.

For low velocities ( St =10.8) the signal is a pure sinus, while at high velocity ( St = 3.5) the signal contains a high noise level. This behavior was also noticed for the NACA 0018. Probably the boundary layer becomes more irregular at Strouhal numbers lower than St < 4.9. Remarkable that at these numbers a peak of the pressure coefficient is found.


126

J.3 Data signal in cavity

Finally, the data signal of the pressure transducer in the cavity is considered. In figure 75 the data signals are given for three Strouhal numbers, (a) St = 3.5, (b) St = 4.9, and (c) St = 10.8. Both the response signal, )(tf , and output signal of the lock-in method, )(tg ,

are shown.

0 50 100 150 200-2

-1

0

1

2cavity; St = 3.5

no. data point [#]

am

plit

ude [

mV

]


output signal, g(t)

0 50 100 150 200 250-0.8

-0.4

-0

0.4

0.8cavity; St = 4.9

no. data point [#]

am

plit

ude [

mV

]


output signal, g(t)

(a) (b)

0 50 100 150 200-0.8

-0.4

0

0.4

0.8cavity, St = 10.8

no. data point [#]

am

plit

ud

e [

mV

]


output signal, g(t)

(c)

Figure 75 Data signals of the pressure transducer in the CAVITY for three different Strouhal numbers,

(a) St = 3.5, (b) St = 4.9 and (c) St = 10.8. A detail of the response signal, )(tf , and output signal of

the lock-in, )(tg , is. The frequency of the speakers is set at 337 Hz and the amplitude of the acoustic

pressure is | p ’| ~ 126 Pa.

A large difference in noise is observed in the signals. For low Strouhal numbers ( St = 3.5 and St = 4.9) the response signals contain a high level of perturbations. The noise level at St = 3.5 is roughly two times as high compared to St = 4.9. At St = 10.8 the response signal is rather smooth again (noise level is approximately 10 times lower compared to St = 3.5).


127

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