Dynamics of turbine flow meters - Pure - Aanmeldenα angle of attack αd damping coefﬁcient m−1 β angle of rotor blade with resect to the rotor axis βav average of the angle

Dynamics of turbine flow meters

Citation for published version (APA):Stoltenkamp, P. W. (2007). Dynamics of turbine flow meters. Eindhoven: Technische Universiteit Eindhoven.https://doi.org/10.6100/IR621983

DOI:10.6100/IR621983

Document status and date:Published: 01/01/2007

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https://doi.org/10.6100/IR621983

https://doi.org/10.6100/IR621983

https://research.tue.nl/en/publications/dynamics-of-turbine-flow-meters(ec509f07-b740-4b93-8eed-8003406f43b4).html

Dynamics ofturbine flow meters

Copyright c©2007 P.W. StoltenkampCover design by Oranje vormgeversPrinted by Universiteitsdrukkerij TU EindhovenCIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Stoltenkamp, P.W.

Dynamics of turbine flow meters / by Petra Wilhelmina Stoltenkamp. -Eindhoven : Technische Universiteit Eindhoven, 2007. - Proefschrift.ISBN 978-90-386-2192-0

NUR 924Trefwoorden: stromingsleer / pulserende stromingen / debietmeters / meetfoutenSubject headings: flow of gases / volume flow measurements / turbine flow meters /pulsatile flow / systematic errors

Dynamics ofturbine flow meters

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Eindhoven

op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn,voor een commissie aangewezen door het College voor Promoties

in het openbaar te verdedigen opmaandag 26 februari 2007 om 16.00 uur

door

Petra Wilhelmina Stoltenkamp

geboren te Heino

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. A. Hirschbergenprof.dr.ir. H.W.M. Hoeijmakers

This research was financed by the Technology Foundation STW,grant ESF.5645

Contents

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . viii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 General description of a gas turbine flow meter . . . . . . . . . . . 11.3 Ideal rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Parameter description . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Reynolds dependency of turbine flow meter readings . . . . . . . . 61.6 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Turbine flow meters in steady flow . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Theoretical models of turbine flow meters . . . . . . . . . . . . . . 10

2.2.1 Momentum approach . . . . . . . . . . . . . . . . . . . . . 102.2.2 Airfoil approach . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Equation of motion . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Effect of non-uniform flow . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Boundary layer flow . . . . . . . . . . . . . . . . . . . . . 172.3.2 Velocity profile measurements . . . . . . . . . . . . . . . . 202.3.3 Fully turbulent velocity profile in concentric annuli . . . . . 212.3.4 Comparison of the different velocity profiles . . . . . . . . 232.3.5 Effect of inflow velocity profile on the rotation . . . . . . . 24

2.4 Wake behind the rotor blades . . . . . . . . . . . . . . . . . . . . . 252.4.1 Wind tunnel experiments . . . . . . . . . . . . . . . . . . . 282.4.2 Effect of wake on the rotation . . . . . . . . . . . . . . . . 29

2.5 Friction forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.1 Boundary layer on rotor blades . . . . . . . . . . . . . . . . 322.5.2 Friction force on the hub . . . . . . . . . . . . . . . . . . . 33

vi

2.5.3 Tip clearance . . . . . . . . . . . . . . . . . . . . . . . . . 352.5.4 Mechanical friction . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Prediction of the Reynolds number dependence in steady flow . . . 382.6.1 Turbine meter 1 . . . . . . . . . . . . . . . . . . . . . . . . 392.6.2 Turbine meter 2 . . . . . . . . . . . . . . . . . . . . . . . . 422.6.3 Effect of tip clearance . . . . . . . . . . . . . . . . . . . . 45

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3. Response of the turbine flow meter on pulsations with main flow . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 A basic quasi-steady model: A 2-dimensional quasi-steadymodel for a rotor with infinitesimally thin blades in incom-pressible flow . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2 Practical definition of pulsation error . . . . . . . . . . . . 523.3 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Determination of the amplitude of the velocity pulsations at the loca-

tion of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Acoustic model . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Synchronous detection . . . . . . . . . . . . . . . . . . . . 593.4.3 Verification of the acoustic model . . . . . . . . . . . . . . 593.4.4 Measurements of velocity pulsation in the field . . . . . . . 64

3.5 Determination of the measurement error of the turbine meter . . . . 653.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.1 Dependence on Strouhal number . . . . . . . . . . . . . . . 693.6.2 Dependence on Reynolds number . . . . . . . . . . . . . . 723.6.3 High relative acoustic amplitudes . . . . . . . . . . . . . . 733.6.4 Influence of the shape of the rotor blades . . . . . . . . . . 74

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4. Ghost counts caused by pulsations without main flow. . . . . . . 794.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Onset of ghost counts . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 Theoretical modelling of ghost counts . . . . . . . . . . . . 804.2.2 Experimental setup for ghost counts . . . . . . . . . . . . . 874.2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 894.2.4 Comparing measurements with results of the theory . . . . . 91

4.3 Influence of vibrations and rotor asymmetry . . . . . . . . . . . . . 934.3.1 Vibration and friction . . . . . . . . . . . . . . . . . . . . . 934.3.2 Rotor blades with chamfered leading edge . . . . . . . . . . 93

vii

4.4 Flow around the edge of a blade . . . . . . . . . . . . . . . . . . . 944.4.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . 944.4.2 Experimental set up for flow around an edge . . . . . . . . 954.4.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . 984.4.4 Comparing measurements with results of the numerical sim-

ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Stationary flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3 Main flow with pulsations . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Pulsations without main flow . . . . . . . . . . . . . . . . . . . . . 1115.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Appendix 113

A. Mach number effect in temperature measurements . . . . . . . . 115

B. Boundary layer theory . . . . . . . . . . . . . . . . . . . . 117B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117B.2 Blasius exact solution for boundary layer on a flat plate . . . . . . . 119B.3 The Von Karman integral momentum equation . . . . . . . . . . . . 120B.4 Description laminar boundary layer . . . . . . . . . . . . . . . . . 121B.5 Description turbulent boundary layer . . . . . . . . . . . . . . . . . 122

C. Measurements . . . . . . . . . . . . . . . . . . . . . . . 125C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.2 Pulsation frequency of 24Hz . . . . . . . . . . . . . . . . . . . . 126C.3 Pulsation frequency of 69Hz . . . . . . . . . . . . . . . . . . . . 127C.4 Pulsation frequency of 117Hz . . . . . . . . . . . . . . . . . . . . 128C.5 Pulsation frequency of 363Hz . . . . . . . . . . . . . . . . . . . . 129C.6 Pulsation frequency of 730Hz . . . . . . . . . . . . . . . . . . . . 130

D. Force on leading edge . . . . . . . . . . . . . . . . . . . . 131

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 133

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 136

viii

Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . 139

Dankwoord . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . 143

Nomenclature

Roman symbols lowercase

a quadratic fit parameter equation 3.25

c0 speed of sound m s−1

f frequency Hz

hblade height of a rotor blade m

k wave number m−1

m′ mass flow kg s−1

n normal unit vector

n number of blades

p pressure Pa

p′ pressure fluctuations Pa

r radius m

rhub radius of the hub m

rout radius of the outer wall m

rtip radius at the tip of the rotor blade m

s distance between two subsequent rotor blades m

t blade thickness or time m or s

tblade blade thickness m

x

u′ velocity fluctuations m s−1

uac acoustic velocity amplitude m s−1

uin inlet velocity m s−1

umax maximum velocity m s−1

uout outlet velocity m s−1

v velocity vector m s−1

w width m

Roman symbols uppercase

A cross-sectional area m2

B′ total specific enthalpy m2 s−2

D pipe diameter m

E relative deviation from ideal rotation equation 2.14

Epuls relative error caused by periodic pulsations equation 3.11

F bf force imposed on the fluid by the body N

FD drag force N

Fe edge force N

FL lift force N

Irotor moment of inertia of the rotor kg m2

K meter factor m3 rad−1

Lblade chord length of a rotor blade m

Lhub length of the hub in front of the rotor m

Q volume flow m3 s−1

R root-mean-square radius√

r2in+r2

out

2 m

S pitch or area m orm2

xi

T temperature or period of the pulsations K or s−1

Tmech mechanical friction torque kg m2s−2

Tair air friction torque kg m2s−2

T bf torque imposed on the fluid by the body kg m2s−2

Td driving torque kg m2s−2

Tf total friction torque kg m2s−2

U mean velocity in the annulus in front of the rotor m s−1

V volume m3

W width of the rotor m

Greek symbols

α angle of attack ◦

αd damping coefficient m−1

β angle of rotor blade with resect to the rotor axis ◦

βav average of the angle of the rotor blades at the root-mean-square radius◦

δ1 displacement thickness m

δ2 momentum thickness m

Φ complex potential m2s−1

φm mass flow kg s−1

Γ circulation m2s−1

γ Poisson’s ratio

µ dynamic viscosity kg m−1s−1

ν kinematic viscosity m2 s−1

ω rotation speed rad s−1

ωid ideal rotation speed rad s−1

xii

ω0 steady rotation speed without pulsations rad s−1

ρ density kg m−3

ρ′ density fluctuations kg m−3

τ viscous stress tensor kg m−1s−2

τw shear stress at the wall kg m−1s−2

Dimensionless numbers

CD drag coefficient,FD/(12ρu

2A)

C ′D drag coefficient,FD/(

12ρu

2wt)

CL lift coefficient,FL/(12ρu

2A)

He Helmholtz number,fLc0

M Mach number,uc0

Pr Prandtl number,ν/a with a the thermal diffusivity

Re Reynolds number,uLν

Sr Strouhal number,fLu

1

Introduction

1.1 Introduction

In industry axial turbine flow meters are used to measure volume flows of gases andliquids. They are considered reliable flow meters and at suitable conditions can attainhigh accuracies in the order of 0.1% for liquids and 0.25% for gases. An accuracyup to 0.02% can reached for high accuracy meters at ideal flow conditions(Wadlow,1998). Turbine flow meters of different design are used in a broad variety of applica-tions, for example in the chemical, petrochemical, food and aerospace industry. Theinternal diameter of these flow meters can vary from very small, e.g. 6mm, to verylarge, e.g. 760mm.

In the Netherlands gas turbine flow meters are commonly used to measure naturalgas flow. Because the Netherlands transported in 2005 95.2 billionm3 of natural gas,small systematic measurement errors can lead to over- or underestimation of largevolumes of natural gas. This makes the accuracy of flow meters crucial atall flowconditions. A new development is the exploration of the possibility to correct flowmeasurements for non-ideal flow conditions on the basis of a physical model for theresponse of the meter to deviations from the ideal flow conditions.

1.2 General description of a gas turbine flow meter

A schematic drawing of a typical turbine flow meter is shown in figure 1.1. In thisdrawing the most important elements of a turbine flow meter are given. Turbineflowmeters are placed in line with the flow. Sometimes they are placed in measuring man-ifolds, where several flow meters are placed in parallel streams, in orderto increasethe overall dynamic range of the set up. Usually the flow passes first through a flow

2 1. Introduction

A B

C

Figure 1.1: Schematic drawing of a turbine flow meter with A) flow straightener and B) rotor.C) shows the position of the mechanical counter

straightener or a flow conditioning plate (A) to remove swirl and create a uniformflow. Subsequently, the flow is forced through an annular channel andthrough the ro-tor (B), see also figure 1.2. The blades of the rotor are often flat plates or have a helicalshape. The shaft and bearings are placed inside the core, which usually is suspended

Figure 1.2: Photograph of the rotor of turbine flowmeter, Instromet type SM-RI-X G250.

downstream of the rotor. There are several ways to detect the rotation speed of therotor. The most common detection methods are mechanical detection and magneticdetection. Mechanical detection of the rotor speed is measured by transferring therotor speed through the rotor axis and via gears to a mechanical counter (C). During

1.2. General description of a gas turbine flow meter 3

magnetic detection a pulse is measured by disrupting a magnetic field every time adesignated point on the rotor, for example the rotor blades, passes a measuring point.These pulses can be processed electronically.

The experiments in this thesis are performed on gas turbine flow meters of Elster-Instromet. The dynamical response measurements have been carried outat the Eind-hoven University of Technology with the gas turbine meter type SM-RI-X G250, seefigure 1.3. This meter has an internal pipe diameter of 100mm. The accuracy of the

Figure 1.3: Photograph of the SM-RI-X G250 turbine flow meter(by courtesy of Elster-Instromet).

flow measurement is 0.1% for volume flows in the range from 20 to 400m3/h. Themeter is designed for pressures ranging from atmospheric pressure upto 20bar (thistype of meter is also available for work pressures up to 100bar). The rotor is madeof aluminium and has helical shaped blades (see figure 1.2). We will referto this me-ter as turbine meter 1. Additional steady flow experiments have been performed byElster-Instromet with simplified prototypes which we refer to as turbine meter 2,3, 4and 5. Additional experiments with oscillatory flow have been performed by Gasuniewith a larger version of the SM-RI-X G250, the SM-RI-X G2500 with a internal pipediameter of 300mm.

4 1. Introduction

w ru i n

w r

u o u t , x = u i n

bu r e l

b

Figure 1.4: Steady flow entering and leaving the rotor for an ideal frictionless rotor withinfinitesimally thin helical rotor blades with blade angleβ.

1.3 Ideal rotation

When ideal rotation is considered, it is assumed that the flow through the turbinemeter is uniform, incompressible and steady, that the rotor rotates with no frictionand that the rotor is shaped as a perfect helix with infinitesimally thin blades. Underthese circumstances the rotation speed of the rotor is determined by thepitch of therotor,S, defined by:

S =2πr

tanβ, (1.1)

with r the radius of the rotor andβ the angle of the rotor blades with respect tothe rotor axis (see figure 1.4). In an ideal case the pitch corresponds tothe axialdisplacement of the fluid during one revolution of the rotor. For a perfecthelicoidalrotor the pitch,S, is constant over the whole radius of the rotor, while the blade angle,β, changes. Because friction is not considered, the flow entering and leaving the rotoris parallel to the blades of the rotor. This means that the inlet velocity and the rotationvelocity are related through the angle of the rotor blades,β, as:

ωidr

uin= tanβ , (1.2)

1.4. Parameter description 5

with ωid the angular velocity of the rotor for the ideal situation considered anduin isthe velocity of the flow entering the rotor. The angular velocity in this ideal situationis

ωid =uin tanβ

r=

2πuin

S. (1.3)

Because the volume flow,Q, is equal to the inflow velocity multiplied by the cross-sectional area of the rotor, i.e.Q = uinA, we find a relationship between the volumeflow and the rotational speed:

Q =AS

2πωid . (1.4)

This relationship is applied in an actual turbine flow meter in the form:

Q = Kωid , (1.5)

whereK is called themeter factor, which is determined by calibration. Ideally,Kshould be a constant.

1.4 Parameter description

In principle for steady flow the meter factorK of a specific meter depends on dimen-sionless parameters such as:

• the Reynolds number Re= uinLν

• the Mach number M= uin

c0

• the ratio of mechanical friction torque,Tmech, to the driving fluid torqueTmech

R3ρu2in

whereL is a characteristic length such as the blade chord length,ν is the kinematicviscosity of the fluid,c0 is the speed of soundR is the root mean square radius ofthe rotor andρ the fluid density. The manufacturer uses steady flow calibrations atdifferent pressures to distinguish between Reynolds number effects and the influenceof mechanical friction. In general the Mach number dependency is a smallcorrectiondue to a Mach number effect in the temperature measurements at high flow rates (seeAppendix A).

In this thesis we will consider unsteady flow. In such case the response of themeter will also depend on:

• the Strouhal number Sr= fLuin

6 1. Introduction

• the amplitude of the perturbations|u′

in|uin

• The ratio of fluid density,ρ, and rotor material density,ρm, i.e. ρρm

wheref is the characteristic frequency of flow perturbations and|u′in| is the ampli-tude of the perturbations.

1.5 Reynolds dependency of turbine flow meter readings

In the ideal case the rotational velocity changes linearly with the volume flow. Inreality friction forces and drag forces cause the rotor to rotate at a rotation speed thatdiffers from the rotational speed of the ideal rotor. The difference between the actualrotor speed and the ideal rotor speed is known as rotor slip. Because thedrag forcesdepend on flow velocity and the viscosity of the medium, therotor slip depends onReynolds number, Re. A meter designer tries to make the volume flow measured bythe meter to be a function that is as linear as possible in terms of the rotational speedfor a dynamic range of at least 10:1. With every meter the manufacturer provides acalibration, that gives the rotor slip as function of the Reynolds number or sometimesas function of the volume flow. This calibration is unique for every meter due tothesensitivity of the meter to small manufacturing differences or differences caused bydamage or wear. One of the aims of the designer is to reduce this sensitivity ofthemeter factor, i.e. the quantityK, for manufacturing inaccuracies, damage or wear.

1.6 Thesis overview

In this thesis, the behaviour of turbine flow meters is investigated experimentallyaim-ing at development of physical models allowing corrections for deviations from idealflow.

In chapter 2 theReynolds number dependenceof the turbine flow meter is inves-tigated analytically. The driving torque on the rotor is obtained by using conservationof momentum on a two-dimensional cascade of rotor blades. Using the equation ofmotion of the rotor, its rotation speed is determined. We use in this chapter a the-oretical model developed by Bergervoet (2005) which we extent by considering theinfluences of non-uniform flow and drag forces. The effect of the inlet velocity profileis investigated using models and measurements. The effect of several friction forcesis modelled analytically. The last part of this chapter compares the model with cali-bration measurements obtained by Elster-Instromet for several turbine flow meters.

Chapter 3 studies theeffect of pulsationssuperimposed on main flow. Pulsationcan induce large systematic errors during measurements. A simplified quasi-steady

1.6. Thesis overview 7

theory predicting these errors, is discussed. Measurements are performed to investi-gate the applicability of this model. A detailed description is given of the measure-ment set up and measurements methods. Finally, the results are discussed.

Chapter 4 deals with the extreem case of chapter 3, where the flow is purelyoscillatory and there is no main flow. This can induce the rotor to rotate and measurea flow while there is no net flow. We call thisghost countsor spurious counts. Thefirst part of this chapter describes two physical models to predict the onset of ghostcounts. The models are compared with experiments. The second part of thischapterinvestigates the flow around the edge of a rotor blade in pulsating flow. First,thisinvestigation is carried out experimentally. These results are compared with adiscretevortex model. The main results of these thesis are summarised in chapter 5.

8 1. Introduction

2

Turbine flow meters in steady flow

2.1 Introduction

In this chapter a model is developed to predict the response of a turbine flow meterin steady flow. The development of a theoretical model describing the behaviour of aturbine flow meter has been endeavoured experimentally and analytically fora longtime (Baker (2000), Wadlow (1998), Lee and Evans (1965), Lee and Karlby (1960),Rubin et al. (1965) and Thompson and Grey (1970)). More recent attempts to un-derstand the behaviour of turbine flow meters use numerical methods to compute theflow field in a turbine flow meter (von Lavante et al. (2003), Merzkirch (2005)). Atheoretical model allows the investigation of, for example, meter geometry, making itpossible to develop better design criteria, or to assess the influence of different fluidproperties. Rather than considering a numerical method we will consider anexten-sion of the more global analytical model as proposed by Thompson and Grey (1970).Our global model aims at understanding important phenomena in the behaviour ofturbine flow meters. Since in practice deviations in the dependence on Reynoldsnumber of 0.2% are significant, we do not expect to succeed in making suchaccuratepredictions of the deviations. We try to obtain some insight into the problem of thedesign of a flowmeter.

The turbine meter is modelled using the equation of motion for the rotor. Theflow passing through the rotor induces a driving torque,Td, on the rotor. First, twoapproaches to obtain this driving torque will be discussed. Next, the influence ofthe inlet velocity,uin at the front plane of the rotor will be investigated by using aboundary layer description, actual velocity measurements in a dummy of a turbineflow meter and a model for fully developed flow. Wind tunnel measurements havebeen performed to investigate the drag forces on the rotor blade. The effect of other

10 2. Turbine flow meters in steady flow

friction forces on the rotor is described and discussed in the following section andtheir individual effect on the rotation speed of the rotor will be shown. Inthe lastpart of this chapter the model is applied to different turbine flow meters at differentReynolds numbers and the results are compared to calibration measurements pro-vided by Elster-Instromet.

2.2 Theoretical models of turbine flow meters

In general two approaches have been used in literature; the momentum approach(Wadlow, 1998) and the airfoil approach (Rubin et al., 1965).

In the momentum approach the integral momentum equation is used to calculatethe driving torque on the rotor. One of the main limitations of this method is thatfullfluid guidanceis assumed. It is assumed that there is a uniform flow tangential tothe rotor blades at the rotor outlet. This assumption is only true for rotors with highsolidity. This implies a gap between successive blades, which is narrow compared tothe blade chord length. Weinig (1964) showed, using potential flow theoryfor a two-dimensional planar cascade, that the ratio of the gap between the blades and bladelength (chord),s/Lblade should be smaller than 0.7 to allow such an assumption.

The airfoil approach on the other hand derives the driving torque on the rotorby using airfoil theory to obtain the lift coefficient of an isolated rotor blade. Withthis approach there is no assumption of full fluid guidance, but blade interferenceis ignored. This means that increasing the number of blades would always increasethe lift force proportionally. Thompson and Grey (1970) improved this approach byusing the two-dimensional planar cascade theory of Weinig (1964) to account for theinterference effects.

Both the integral momentum method and the airfoil method will be explained inmore detail in the following sections. We later actually use only the integral momen-tum method, which has been used earlier in simplified form by Bergervoet (2005) atElster-Instromet.

2.2.1 Momentum approach

The turbine meter is a complex three-dimensional flow device (see figure 2.1). As anapproximation this three-dimensional problem will be treated as a two-dimensionalinfinite cascade of rotor blades with uniform axial flow,uin, at radiusr as approxima-tion of the flow inside an annulus betweenr andr+ dr. The x-direction refers to theaxial direction. The y-direction refers to the azimuthal direction (see figure 2.2). Theradial velocity is neglected and constant rotation with a rotational angular velocity ωis assumed. To obtain the torque on the rotor we will integrate over the blade lengthin radial direction. The control volume enclosing the rotor is shown in figure2.2.

2.2. Theoretical models of turbine flow meters 11

d rr t i p

r h u bw

h b l a d e

rq

x

Figure 2.1: The rotor of the turbine flow meter. We assume thatthe flow in an annulus be-tweenr and r + dr behaves as the flow in a two-dimensional infinitely longcascade shown in figure 2.2.

To calculate the driving torque on the rotor, the integral mass conservationlaw andintegral momentum equation is used for this two-dimensional cascade of blades:

ddt

∫∫∫

CV

ρdV +

∫∫

CS

ρv · ndA = 0 , (2.1)

ddt

∫∫∫

CV

ρvdV +

∫∫

CS

ρv (v · n) dA = −∫∫

CS

pndA+

∫∫

CS

τndA+ F bf ,(2.2)

applied to a fixed control surfaceCS enclosing the rotor, this surface has an outernormaln, the fixed control volume withinCS is denoted asCV , ρ is the fluid density,v is the velocity vector,p is the pressure,τ is the viscous stress tensor andF bf arethe forces imposed on the fluid by the turbine.

Full fluid guidance is assumed; the flow leaves the rotor with a velocity paralleltothe blades along the whole circumference (or the y-direction in our 2D model,figure2.2). This implies that we neglect radial velocities and the effect of the Coriolisforces. We assume that the flow enters the rotor without any azimuthal velocity, vθ =0 (in a two dimensional representationvy = 0). Assuming steady incompressibleflow and applying the conservation of mass (equation 2.1) to a volume element ofheight dr (figure 2.1), we get:

uin,xdAin = uout,xdAout , (2.3)


w ru i n

u o u t , y

w r

u o u t , x = u i n

b

u r e l

b

C S

xy

n

n

t

L b l a d e

u r e l

W

Figure 2.2: Flow entering and leaving the cascade representing the rotor in an annulus be-tweenr andr + dr.

whereuin,x anduout,x are x-component of the the incoming and outgoing velocity,respectively, and dAin and dAout are the inflow area and the outflow area, respec-tively. If the inflow and outflow area are assumed to be equal and the flow isincom-pressible, dAin = dAuit = 2πrdr, so that the x-component of the incoming velocityis equal to the x-component of the outgoing velocity, i.e.uin,x = uout,x.

Using the same assumptions as mentioned above and neglecting the viscousforces, Re>> 1, the momentum equation in the y-direction for a steady flow throughan element dr becomes:

ρ ((uout,y + ωr)uout,xdAout − uin,xωrdAin) = dFbf,y , (2.4)

From the velocity diagram in figure 2.2 it can be seen that:

uout,y = uout,x tanβ − ωr . (2.5)


Substituting equations 2.3 and 2.5 in equation 2.4, the y-component of the forceimposed by the rotor on the fluid, dFbf,y is found:

dFbf,y = ρu2out,x tanβdAout − uin,xωrdAin . (2.6)

The force of the fluid on the rotor is opposite and equal to the force of the rotor on thefluid, dFbf,y = −dFfb,y. The torque exerted by the fluid element on the rotor axis,dTd, is estimated to be:

dTd = rdFfb,y . (2.7)

By integrating this equation from the radius of the rotor hub,rhub to the rotor tip,rtip(see figure 2.1), the driving torque on the rotor is:

Td = −∫ rtip

rhub

ρu2out,x(tanβ)rdAout +

∫ rtip

rhub

ρuin,xωr2dAin . (2.8)

2.2.2 Airfoil approach

An alternative method to obtain the driving torque on the rotor, is the airfoil approach.Again the element of the rotor at radiusr and thickness dr is approximated as aninfinite two-dimensional cascade of rotor blades (see figure 2.3). In contrast to themomentum approach there is no assumption that flow is attached. The driving torqueon the rotor blade is now evaluated by determining the lift and drag forces onthe rotorblades in a coordinate system fixed to the blade. The lift force,FL, acts perpendicularto the relative inlet velocity,uin,rel = (uin,x, ωr), and the drag force,FD acts parallelto this inlet velocity. The y-component of the force of the flow on the blade can nowbe expressed in terms of lift,FL, and drag,FD;

Fy = n (−FL cosφ+ FD sinφ) , (2.9)

whereφ = β−α = arctan(

ωruin,x

)

, with β the angle of the rotor blade (with respect

to the x-axis),n is the number of blades andα the angle of attack of the incomingflow. The lift- and drag coefficient are defined as:

CL =FL

12ρu

2in,relLblade

,

CD =FD

12ρu

2in,relLblade

,(2.10)

whereLblade is the chord of the blade. The lift and drag coefficients are functionsof the angle of attack,α, depend weakly on Reynolds number and on Mach number.


b

xy

t

L b l a d e

u i n , r e lF L

F D

F L , y

F D , y

a f

w ru i n

f

Figure 2.3: Lift and drag force acting on a blade of a two dimensional cascade

Using these coefficients the driving torque on a rotor withn blades can be written as:

Td =

∫ rtip

rhub

1

2nρu2

in,relLblade (−CL cosφ+ CD sinφ) rdr . (2.11)

2.2.3 Equation of motion

The driving torque,Td, is known from equation 2.8 or 2.11. To determine the angularvelocity,ω, of the rotor, the equation of motion of the rotor is used:

Irotordωdt

= Td − Tf , (2.12)

whereIrotor is the moment of inertia of the rotor andTf is the friction torque onthe rotor, assuming a quasi-steady flow through the rotor. Using equation 2.8 or 2.11


for the torque implies that we assume a quasi-steady flow through the rotor. In thischapter we investigate the rotor in steady rotation, for which the equation of motionreduces to:

Td = Tf . (2.13)

The different friction forces will be discussed in the following sections. This equationcan be used to predict the steady rotation speed of the rotor,ω. By comparing thisrotation speed with the ideal rotation speed,ωid (see equation 1.3), the deviation ofthe rotation speed of the turbine meter from ideal rotation can be determined as:

E =ω − ωid

ωid

. (2.14)

Calculating the deviation at various Reynolds numbers, Re, the dependence of a tur-bine meter can be estimated.

In the following sections the analysis will be applied using the momentum ap-proach (equation 2.8) to two types of turbine flow meters. The first one, referred toas turbine meter 1, is the Instromet SM-RI-X G250 with a diameter ofD = 0.1 mused in the experiments at the set up in Eindhoven. The second one is a simplifiedturbine meter with diameter ofD = 0.2 m, this rotor will be referred to as turbinemeter 2. The second turbine meter has a simplified geometry. An example of thissimplification is the geometry at the rotor tip (see section 2.5.3). This simplified ge-ometry should allow a better comparison of experiment with the theory. Informationabout the geometry of the two flow meters is given in table 2.1 The chord length ofthe rotor blades of turbine meter 1 can be calculated using:

Lblade(r) =W

cosβ(r), (2.15)

with β = arctan(

2πrS

)

the angle of the blade relative to the rotor axis. The bladesof the second turbine meter, turbine meter 2, are reduced at the tip to a chordlengthof Lblade(rtip) = 0.035m. The chord length of the rotor blades of this turbine metercan be written as:

Lblade(r) = Lblade(rhub) +Lblade(rtip) − Lblade(rhub)

hblade

(r − rhub) . (2.16)

In the following sections the effect of non-uniform flow, the blade drag and otherfriction forces are investigated separately, the deviation from the ideal rotation iscalculated for several flows up toQmax as indicated for the meter. Two scenarioswere followed; in the first scenario the calculations were done using the propertiesof air at 1 bar (absolute pressure),ρ = 1.2 kg/m3 andν = 1.5 × 10−5 m2/s, and


turbine meter 1 turbine meter 2

pipe diameter,D (m) 0.1034 0.2030blade thickness,t (mm) 1.6 4

number of blades,n 16 14rhub/D 0.360 0.250rout/D 0.500 0.500S/D 2.704 3.941W/D 0.213 0.148

hblade/D = (rtip − rhub)/D 0.140 0.240Lhub/D 0.763 1.049

Table 2.1: Dimensions of the two turbine flow meters used in the calculations, whererhub isthe radius of the hub,rout is the radius of the outer wall,rtip is the radius at thetip of the blades,S is the pitch (equation 1.1),W is the width of the rotor,hblade

is the height of the blade (span of the blades) andLhub is the length of the hub infront of the rotor. Except for the blade thicknesst and the number of bladesn, allvalues are made dimensionless with the diameter,D.

in the second scenario the properties of natural gas at 9 bar (absolute pressure) wereused,ρ = 7.2 kg/m3 andν = 1.5 × 10−6 m2/s. These conditions correspond tothe test conditions used by Elster-Instromet. The resulting deviation,E, is plottedagainst the Reynolds number, Re= ULblade/ν, whereLblade is the length of a rotorblade measured at the tip andU the velocity at the rotor.

For the calculation in this chapter only the momentum approach is being used.This approach assumes full fluid guidance, i.e. attached flow. This is a good approxi-mation, if the ratio of the distance between the blades and blade length is sufficientlysmall, s/Lblade < 0.7. In case of the first turbine meter this assumption is valid.For turbine meter 2 this assumption is no longer valid at the tip of the blades. How-ever, the departure from full fluid guidance is expected to be small. Using the theoryof (Weinig, 1964), we estimate that the tangential velocityuout,y will be about 5%smaller than the tangential velocity for full fluid guidance. The reduction in thetan-gential velocity decreases the driving torque exerted by the flow on the rotor and thisdecreases the rotation speed of the rotor. Because this effect will be small in this case,we will ignore it in our model.

2.3 Effect of non-uniform flow

As can be seen from equation 2.8 the driving torque depends on the velocity enteringthe flow meter. The flow entering the turbine is generally non-uniform. Boundary

2.3. Effect of non-uniform flow 17

layers will form along the walls and in pipe systems swirl inevitably occurs duetoupstream bends. Parchen (1993) and Steenbergen (1995) showedthat swirl decaysextremely slowly. Swirl can have effect the accuracy of turbine meters (Merzkirch,2005). Properly designed flow straighteners as designed by Elster-Instroment placedin front of a turbine flow meter reduce the effect of swirl considerably.Therefore inthe calculation we assume that there is no azimuthal velocity (no swirl). We limit ourdiscussion to the non-uniformity of the axial velocity,uin(r).

Thompson and Grey (1970) predicted that the inlet velocity profile plays anim-portant role in the rotation speed of the rotor.

The influence of the velocity profile entering the rotor will be investigated in thissection. The shape of the velocity profile entering the rotor is first calculated usingboundary layer theory. Velocity profile measurements carried out in a dummyof aturbine meter will be compared with the boundary layer theory and a fully developedturbulent annulus flow assumed by Thompson and Grey (1970). The rotation rate ofa rotor for velocity profile based on boundary layer theory and for a measured flowprofile will be compared with predictions of the ideal rotation rate.

2.3.1 Boundary layer flow

The flow enters the turbine meter, passes a flow straightener and continuesthroughan annular pipe segment of lengthLhub around the hub of the turbine meter (see 1.1).Upon entering the annulus, the gas is accelerated because of the area contraction.Due to this acceleration the thickness of the boundary layers is strongly reduced. Atthe leading edge of the hub a new boundary layer starts to form on the hub and on theouter wall. The velocity profile is assumed axisymmetric and can be divided in threeregions (see figure 2.4). The first region is the boundary layer on the hub. The secondregion is the region between the boundary layers, where the velocity is approximatelyuniform. The third region is the boundary layer on the outer pipe wall.

Calculation are carried out for two cases; laminar and turbulent boundary layers.The transition from laminar to a turbulent flow occurs for flat plates under optimalconditions around a Reynolds number of ReLhub

≈ 3×105 (Schlichting, 1979). Thiswould imply that there is a significant laminar part of the boundary layer on thehubeven for ReLhub

> 3 × 105. However, we will assume that above a critical Reynoldsnumber the boundary layer is turbulent from the start, ignoring the effectof transition.

The boundary layer thickness is calculated using the von Karman integral mo-mentum equation (see Schlichting (1979)). Appendix B provides a brief discussionof boundary layer theory. The von Karman equation obtained by integration of themass and momentum equations over the boundary layer is:

ddx

(

U2δ2)

+ δ1UdUdx

=τwρ, (2.17)


I

I I

I I I

u ( r )

d ( x ) r h u b

h b l a d e

rotor

straig

htener

h u bL h u b

o u t e r w a l l

xr

W

r o u t

Figure 2.4: The three different regions of the velocity profile in the turbine meter

with U the velocity outside the boundary layers,δ1 the displacement thickness (fordefinition see equation B.3),δ2 the momentum thickness (for definition see equationB.4) andτw the shear stress at the wall. For the calculation of the laminar boundarylayer, a third order polynomial description of the boundary layer profile isused incombination with Newton’s law forτw (see Appendix B). This was found to be anaccurate description of a laminar boundary layer by Pelorson et al. (1994) and Hof-mans (1998). For turbulent flow the boundary layer is described using a1/7th powerlaw description for the velocity profile combined with the empirical law of Blasiusfor the wall shear stress (see Appendix B). Using these models, the displacementthickness,δ1, the momentum thickness,δ2, and the shear stress at the wall,τw, arecalculated just upstream of the turbine flow meter. The mean velocity in the annulus,U , is corrected for the boundary layer on the hub as well as on the pipe wall.Usingthe definition of displacement thickness,δ1, this velocity can be written as:

U(x;Q, δ(x)) =Q

π ((rout − δ1)2 − (rhub + δ1)2), (2.18)

whereQ is the volume flow,rout is the radius of the outer wall andrhub is the radiusof the hub. The boundary layers on the outer pipe wall and on the hub areassumedto have the same thickness.

The velocity profile in front of the rotor of a turbine meter with geometrical di-mensions equal to the turbine meter 1, is calculated. This meter has a radius of theouter wall,rout = 0.050 m and a radius of the hubrhub = 0.037 m. The hub lengthin front of the rotor isLhub = 0.076 m. For laminar boundary layers figure 2.5(a)shows the calculated velocity profile in the annulus just upstream of the rotor. For tur-bulent boundary layers the velocity profile is plotted in figure 2.5(b). As expected the


0.7 0.75 0.8 0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

r/rout

u/u m

ax

ReL

hub

= 1.4 X 105Re

Lhub

= 3.9 X 104Re

Lhub

= 1.2 X 104Re

Lhub

= 3 X103

rhub

(a) laminar boundary layers

0.7 0.75 0.8 0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

r/rout

u/u m

ax

ReL

hub

= 3 X103

ReL

hub

= 1.2 X 104

ReL

hub

= 3.9 X 104

ReL

hub

= 1.4 X 105

rhub

(b) turbulent boundary layers

Figure 2.5: Velocity profile entering the rotor for turbine meter 1 with a diameter 0.1034mcalculated using boundary layer theory. The velocity,u, divided by the maxi-mum velocity,umax is plotted against the radius for different Reynolds numbers(Re= ULhub/ν = 3×103, 1.2×104, 3.9×104 and1.4×105. (a) shows the ve-locity profile with laminar boundary layers, (b) the velocity profile with turbulentboundary layers.


laminar boundary layers are thinner than the turbulent boundary layers.The velocityprofile for turbulent boundary layers is more uniform than that for laminarflow.

2.3.2 Velocity profile measurements

To examine whether the boundary layer description of the velocity profile is an ad-equate approximation of the velocity profile, measurement were carried outwith ahot wire anemometer and a Pitot tube in the set up described in section 3.3. In thisset up turbine flow meter 1 with a diameter ofD = 0.1 m, is placed at the end ofa pipe with a length of more than 30 times its diameter. The pipe flow is suppliedby a high pressure dry air reservoir (60bar). A choked valve is controlling the massflow through the pipe. In order to measure the velocity profile just upstreamof therotor, the flow meter was replaced by a dummy. The dummy is a replica of the for-ward part of the meter, including the flow straightener, up to the rotor. The remainderof the flow meter, including the rotor, has been removed providing easy access forthe measurement probes. The Pitot tube has a diameter of 1mm and is connectedto an electronic manometer, Datametrics Dresser 1400, and a data acquisition PC.The single wire hot wire anemometer (Dantec type 55P11 wire with 55H20 support)is also connected to a PC. More details of the set up can be found in section 3.3.The pressure and velocity are determined by averaging over a 10s measurement ata sample frequency offs = 10 kHz. Before measuring the velocity profile just infront the rotor (but in absence of the rotor), the velocity profile in the pipeupstreamof the turbine flow meter was measured using the Pitot tube. Measurements wereperformed at four different velocities in the pipe, 2, 4, 10, 15m/s. The measuredprofiles are plotted in figure 2.6. The Reynolds number, ReD, mentioned in figure 2.6is based on the diameter,D, of the pipe and the maximum velocity measured,umax.The measured velocity profile is symmetric and approaches that of a fully developedturbulent pipe flow.

Measurements of the annular flow 1mm downstream of the dummy of the for-ward part of the meter were performed at seven different average velocities in the pipe(0.5, 1, 1.5, 2, 4, 10 and 15m/s), resulting in Reynolds numbers, Re= ULhub/ν,whereLhub is the length of the hub in front of the rotor (see figure 2.4) andUthe mean velocity in the annulus outside the boundary layers (equation 2.18).ThisReynolds number ranges from3.0 × 103 up to1.5 × 105. From the measurementsshown in figure 2.7, it can be seen that the velocity profile is asymmetric. Theasymmetry is increasing with increasing Reynolds number. It has a maximum ve-locity closer to the outer wall than to the hub. For lower Reynolds numbers nearthe walls the velocity profile resembles the laminar boundary layer velocity profile,for Reynolds number above104 the velocity profile resembles more the turbulent


−0.5 0 0.50.40.30.20.1−0.1−0.2−0.3−0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/D

u/u m

ax

ReD = 1.5 X 104

ReD = 2.8 X 104

ReD = 7.1 X 104

ReD = 1.1 X 105

Figure 2.6: The velocity profile in the pipe just upstream of the turbine flow meter, measuredat four different Reynolds numbers, ReD = umaxD/ν = 1.5 × 104, 2.8 × 104,7.1 × 104 and1.1 × 105.

boundary layer profile.It is difficult to determine the exact velocity profile near the wall of the pipe and

the hub. This can be seen in figure 2.7. The velocity is measured 1mm downstreamof the dummy of the turbine meter. At this point there is a flow forr/rout > 1,because of entrainment of air in the airjet flowing out of the dummy (figure 2.9). Wetherefore observe some velocity at the location of the pipe wall,r/rout = 1, wherein the pipe the velocity vanishes.

2.3.3 Fully turbulent velocity profile in concentric annuli

Fully developed turbulent axisymmetric axial flow in a concentric annulus hasbeenstudied in literature, because of the many engineering applications and in order toobtain fundamental insight in turbulence. Brighton and Jones (1964) found experi-mentally that the position of the maximum velocity of such fully developed flows iscloser to the inner wall than to the outer pipe wall. The position depends on Reynoldsnumber and ratiorhub/rout of the inner wall radius,rhub, and the outer wall radius,rout. The results found here differs in that respect.


0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.050

0.2

0.4

0.6

0.8

1

r/rout

u/u m

ax

Re = 3.1 X 103

Re = 5.6 X 103

Re = 1.2 X 104

routr

hub

(a) Re< 2 × 104

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.050

0.2

0.4

0.6

0.8

1

r/rout

u/u m

ax

Re = 2.0 X 104

Re = 3.9 X 104

Re = 9.2 X 104

Re = 1.4 X 105

routr

hub

(b) Re≥ 2 × 104

Figure 2.7: Velocity profile at the entrance of the rotor (turbine meter 1,D = 0.1034 m)measured with the hot wire anemometer 1mm downstream of a dummy of theforward part of the meter. The velocity,u, normalised by the maximum velocity,umax as a function of the radius for four different Reynolds numbers.


u ( r )

r h u b

straig

htener

h u bL h u b

o u t e r w a l l

xr

r o u t

1 m m

p r o b e

Figure 2.8: Schematic drawing of the position of the hot wireduring the velocity measure-ments.

p i p e w a l lf l o w e n t r a i n m e n t

m a i n ( j e t ) f l o w

Figure 2.9: The air outside the pipe is entrained in the airjet exiting the pipe

2.3.4 Comparison of the different velocity profiles

The velocity profile calculated using boundary layer theory (figure 2.5),the mea-sured profile (figure 2.7) and the profile of a fully developed turbulent flow as foundby Brighton and Jones (1964) are quite different. Comparing the result of the bound-ary layer calculations for turbine flow meter 1 with the measurements in the samemeter, the measured profiles show a clear asymmetry dependent on the Reynoldsnumber. Fully developed turbulent flow in an annular channel (e.g. Brighton andJones (1964)) displays a maximum velocity closer to the inner wall than to the outerwall. However, in our measurements the maximum velocity is closer to the outerwall. This indicates that the measured velocity profile does not resemble the fullydeveloped turbulent flow in an annulus. This is not surprising, since the length ofthe hub,Lhub, is relatively short,Lhub ≈ 5.5(rout − rhub). The asymmetry in the


measured profile can be caused by flow separation at the front of the hub, resultingin a velocity profile with higher velocity along the outer wall (see figure 2.10).Theobserved velocity maximum would be due to the flow separation at the sharp edge ofthe nose of the hub. Similar behaviour is observed downstream of a sharpbend in apipe.

r h u b

r o u t

rotor

straig

htener

h u bL h u b

u

o u t e r w a l l

xr

Figure 2.10: Flow is expected to separate at the leading edgeof the hub causing the flow toaccelerate close to the outer wall

2.3.5 Effect of inflow velocity profile on the rotation

To investigate the effect of the inlet velocity profile on the driving torque,Td, thedriving torque is calculated using the predicted velocity profile based on boundarylayer theory. The mechanical friction forces, the fluid friction and the thickness of theblades are ignored. The results are compared to the calculation of the driving torquepredicted for a uniform velocity. As we assume incompressible flow, the continuityequation gives that the incoming velocity is equal to that of the axial component ofthe outgoing velocity,uin = uin,x = uout,x. The momentum equation (equation 2.8)reduces to:

Td = −∫ rtip

rhub

ρuin (uin tanβ + ωr) 2πr2dr . (2.19)

For steady flow and in absence of friction the equation of motion of the rotor (equa-tion 2.13) reduces to:Td = 0. For a given geometry of the rotor and a knownincoming velocity profile, the rotation speed of the rotor can then be calculated. Asthe velocity profile depends on the Reynolds number, Re= ULhub/ν, the deviationof the rotation speed from ideal rotation speed for a uniform inflow,E (see equation2.14), is plotted against Reynolds number. In figure 2.11 the deviation in rotationspeed has been plotted for the laminar and turbulent boundary layer profiles (figure2.5) and for the measured profile (figure 2.7). Compared to a uniform flowthe rota-

2.4. Wake behind the rotor blades 25

103

104

105

106

0

1

2

3

4

5

6

7

8

ReL

hub

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

velocity profile measurementsturbulent boundary layerslaminar boundary layers

Figure 2.11: The deviation of the rotation speed,ω, from the rotation speed for a uniforminlet velocity profile,ωid versus Reynolds number, Re= ULhub/ν. Turbulentboundary layer approximation (solid line), laminar boundary layer approxima-tion (dashed line) and the measured velocity profile (◦).

tion speed of the rotor increases in the order of one percent for a velocity profile basedon laminar or turbulent boundary layer theory. The turbulent boundarylayer causesthe rotor to rotate faster than the laminar boundary layers. The measured velocityprofile induces much larger deviations. As we are aiming for an accuracy of 0.2%, itis clear that the velocity profile plays a very significant role in the rotation speed ofthe rotor, as already observed by Thompson and Grey (1970). In further calculationsdiscussed in this chapter, the boundary layer model is used. We have to keep in mindthat the measured profile induces a larger deviation.

2.4 Wake behind the rotor blades

The flow around the rotor blades does not only provide a driving torque, but the flowalso exerts a drag force on the rotor. The effect of the forces caused by the pressuredifference between the pressure and the suction side of the rotor blade and by thefriction of the fluid on the solid surface of the blades (described in section 2.5.1) canbe included in the momentum conservation balance described in section 2.2.1. To


include the pressure drag, a model for the wake is proposed. In this model we willassume that the wake of the blade in the rotor has the same structure as for a singleisolated blade in free stream (see figure 2.12).

w a k er o t o r b l a d e

u i n , r e l u o u t , r e ln

C S AA

w w a k e

f m

Figure 2.12: Wake behind a rotor blade.

Betz, Prandtl and Tietjens (1934) found that it is possible to calculate the dragforce on a body in an unbounded uniform flow by applying a momentum balance ona large control surface surrounding the body. The control volume is chosen aroundthe rotor blade, with a control surfaceCS with a normal vectorn as shown in figure2.12. The control volume has to be chosen far from the body. There, thestreamlinesin the flow are again approximately parallel and the pressure over the wakecan beconsidered uniform and equal to the pressure of the uniform flow. Therotor blade andthe wake cause a displacement of the flow over the sides. We apply the momentumequation on this controle volume for steady incompressible flow.

Assuming that outside the wake the velocity,u, can be approximated by the freestream velocity,u∞ = uin, this equation reduces to:

FD = ρ

∫

wake

uout (u∞ − uout) dy = ρu2∞δ2,wake , (2.20)

where the integral can be limited to the wake, becauseuout = u∞ outside the wakeand δ2,wake is the momentum thickness of the wake. With this equation the dragcoefficient of a blade of lengthLblade and thicknesst, C ′

D = CDLblade

t= FD

1

2ρu2

in,relt,

can be determined from the velocity distribution in the wake.Note, that if this momentum approach is used for a model of the wake, in which

the velocity directly behind the blades is assumed zero and the pressure in thewakeequal to the pressure of the uniform main flow, the drag coefficient vanishesC ′

D = 0(Prandtl and Tietjens (1934)). This is not a realistic value for the drag coefficient.Obviously, the pressure at the base of the blade is lower than the free stream pressureand a drag is experienced by the blade. The flow just behind the blade is extremely


complex. We will therefore consider the wake at some distance from the trailing edgeof the blade.

t

L b l a d e

U

Figure 2.13: Rounded edge geometry used by Hoerner (1965). This geometry withLblade/t = 6 has a drag coefficient witht as reference length ofC ′

D = 0.64.

Hoerner (1965) (see also Blevins (1992)) found experimentally that a blade witha rounded nose and a squared edged base, with the dimensionsLblade/t = 6 (seefigure 2.13) has a drag coefficientC ′

D = 0.64 for ReLblade> 104. This geometry

is comparable to our rotor blade, except for the geometry of the trailing edge. Theratio of the thickness and the blade length of a rotor blade of turbine meter 1 isLblade/t ≈ 20 and for turbine meter 2 the ratio isLblade/t ≈ 8. The chamfered, sharpedge reduces the drag coefficient, because the flow will not separate immediately atthe edge, which reduces the thickness of the wake. This is illustrated in figure 2.14.

w a k er o t o r b l a d e

u i n , r e l u o u t , r e l

w w a k e

n

C S AA f m

Figure 2.14: Wake behind a rotor blade with chamfered trailing edge.

The drag consist of a combination of the pressure drag and of the drag caused byskin friction. The skin friction will be calculated separately. To determine the effectof the skin friction compared to the pressure drag, the drag coefficient caused by lam-inar and turbulent boundary layers is now estimated by considering the rotor blade asa flat plate. For laminar boundary layers the wall shear stress can be calculated usingBlasius’ numerical result (Schlichting (1979) and Appendix B). The drag coefficient,


C ′D, caused by the skin friction on both side of the blades is:

C ′D,friction =

2∫ Lblade

x=0 τwdx12ρU

2t=

1.328√

ReLblade

Lblade

t(2.21)

For turbulent boundary layers the drag coefficient is found empirically (Schlichting,1979) to be:

C ′D,friction = 0.148 Re

− 1

5

Lblade

Lblade

t(2.22)

For the rotor blades considered in this chapter, the contribution of the skin frictionto the drag coefficient depends on the Reynolds number and whether the boundarylayers are laminar or turbulent. For the range of Reynolds numbers used inthe presentexperiments the contribution to the drag coefficient of the skin friction is typicallyC ′

D ≈ 0.05 for laminar boundary layers andC ′D ≈ 0.25 for turbulent boundary layers

for turbine meter 1. For turbine meter 2 we findC ′D ≈ 0.03 for laminar boundary

layers andC ′D ≈ 0.18 for turbulent boundary layers. As the total dragC ′

D = 0.64 forthe blade geometry with a blunt trailing edge (see figure 2.13 and Hoerner (1965)), weexpect that the contribution of the pressure drag will be in the order of 0.5. Assumingthat the wake has a thickness equal to the blade thickness,wwake = t, and that thevelocity in the wake is half the mainstream velocity,uwake = 1

2uin, using equation2.20, we can calculate that the rotor blade has a drag coefficientC ′

D = 0.5. In caseof the rotor blade with a chamfered trailing edge, we will also assume a wake withavelocityuwake = 1

2uin. The wake thickness,wwake will be tuned in order to matchthe measured values ofC ′

D for a two-dimensional model of the rotor blade. Theexperiments used to measure this drag coefficient are discussed in the next section.

2.4.1 Wind tunnel experiments

In a wind tunnel with a test section of a heighthwt = 0.5 m and widthwwt = 0.5 ma two-dimensional wooden model of a single rotor blade is placed. The blademodelhas a thickness,t, of 1.8cm, a length,Lblade of 14.6cm and a width,wblade of 48.9cm. It has a rounded leading edge and a chamfered trailing edge (see figure 2.15).The angle of the trailing edge is45◦.

The blade is connected to two balances with rods and ropes. The first balanceis a Mettler PW3000 with a range of 3kg and measures the drag force,FD inducedby the flow around the blade. The second one is a Mettler PJ400 with a 1.5kgranges and measures the lift force,FL. Both mass balances have an accuracy of 0.1g. Measurements were carried out for Reynolds numbers, Re= uLblade/ν, basedon the blade length, ranging from ReLblade

= 4 × 104 up to3 × 105 at blade angles,


u

t

L b l a d ew b l a d e

w w t

h w ta

Figure 2.15: Wind tunnel set up.

α from −3◦ to 3◦. The blade angles are determined using an electronic level meter(EMC Paget Trading Ltd model: 216666).

In figure 2.16 the drag coefficient,C ′D = CD

Lblade

t= FD/

(

12ρu

2wbladet)

, isplotted against the Reynolds number, Re= uLblade/ν for a blade angle,α = 0.3◦.The measurements show a drag coefficient,C ′

D, between 0.1 to 0.35, much lowerthanC ′

D = 0.64 found for the similar geometry with blunt trailing edge by Hoerner(1965).

Figure 2.16 also shows the estimated skin friction for laminar and turbulent bound-ary layers for the wind tunnel model. The contribution of the skin friction to thedragcoefficient is significant for turbulent boundary layers.

An other consequence of the asymmetric shape of the chamfered edge of the rotorblade, is that at zero incidence,α = 0, the blade generates a lift force. This can beseen in figure 2.17. This effect has not yet been included in the theory described inthis chapter, because we expect that the lift coefficient of a blade in a cascade stronglydeviates from a single blade in uniform flow as presented here.

In a closed wind tunnel cascade measurements are only possible at0◦ incidence,because the walls prevent deflection of the flow. For measurements at different anglesof incidence a special cascade wind tunnel should be used (Jonker, 1995). Measure-ments obtained for a five blade cascade with typical ratio of distance betweentheblades and blade length,s/Lblade, of 0.55 indicated that the measured drag coeffi-cient,C ′

D, values are close to the value obtained for a single blade.

2.4.2 Effect of wake on the rotation

The model described above is included in the momentum equation. The reduced ve-locity in the wake can be described with the displacement thickness,δ1,wake, and the


0 0.5 1 1.5 2 2.5 3

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

ReL

blade

C’ D

=F

D/(

1/2

ρ u in

,rel

2 w

blad

e t)

turbulent

laminar

Hoerner (1965)

Figure 2.16: The drag coefficient,C ′

D, as a function of Reynolds number, ReLbladefor flat

plate with round nose and45◦ chamfered trailing edge measurements at anangle of attackα = −0.3◦. The arrow indicates the drag coefficient of 0.64found in Hoerner (1965), the dashed line is an approximationfor the part ofthe drag coefficient in case of laminar boundary layers and the solid line is theapproximation for turbulent boundary layer.

−3 −2 −1 0 1 2 3−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

α (o)

CL=

FL/(

1/2

ρ u in

,rel

2 w

blad

e Lbl

ade)

Figure 2.17: The lift coefficient,CL, is plotted at various angles of attack,α, for Reynoldsnumber, ReLblade

> 3 105. The dashed line is a linear fit through the datapoints. We observe a net lift coefficientC ′

L(0◦) = 0.1 at a zero angle of attack,α = 0. This is due to the asymmetry in the blade profile at the trailing edge.

2.5. Friction forces 31

momentum thickness,δ2,wake (see Schlichting (1979) and Appendix B). Applyingmass conservation and using the displacement thickness, we find:

uout,x =1

1 − nδ1,wake

2πr cos β

uin,x , (2.23)

wheren is the number of blades of the rotor. Using both the displacement thick-ness,δ1,wake, the momentum thickness,δ2,wake, and equation 2.23 in the momentumbalance, the driving torque (equation 2.8) becomes:

Td = −∫ rtip

rhub

ρu2

in,x(tanβ)r(

1 − nδ1,wake

2πr cos β

)2

(

2πr − n(δ1,wake + δ2,wake)

cosβ

)

dr

+ 2π

∫ rtip

rhub

ρuin,xωr3dr .

In the proposed model, in which the velocity in the wake, with a wake thicknesswwake, of half of the mainstream velocity,uwake = 1

2uout, the displacement thicknessis δ1,wake = 1

2wwake and the momentum thickness isδ2,wake = 14wwake.

From the wind tunnel measurements described above, it is found that the dragcaused by the wake behind the blade is overestimated by using the drag coefficientin Hoerner (1965) ofC ′

D = 0.64. To account for this, the thickness of the wake canbe changed. If a wake thickness is chosen equal to the blade thickness,wwake = t,the pressure drag of the blunt body is obtained. By reducing the wake thickness, thedrag coefficient of the rotor blade can be reduced to the values obtainedfrom themeasurements. This will be applied in our calculations

Neglecting friction forces and assuming a uniform inflow, the deviation fromtheideal rotor speed caused by different drag coefficients, or different wake thicknesswwake, has been calculated. For the turbine flow meters 1 and 2 the effect of wakethickness can be seen in table 2.2. In this approximation this effect is not dependenton Reynolds number. We observe a significant effect of the drag on thedeviation,E,of the order of 2%.

2.5 Friction forces

Although turbine flow meters are designed to rotate with minimum friction, there areseveral important friction forces that influence the rotation speed of the rotor. Thereare two different kind of friction forces, the mechanical friction force and the frictionforces induced by the flow. Mechanical friction forces are the forcescaused by thebearings and the magnetic pick up placed on the meter. Flow induced friction consistsof the fluid drag on the blades and on the hub, the fluid friction at the tip clearance


thickness of the wake deviation,E = ω−ωid

ωid× 100%

turbine meter 1 turbine meter 2

0 0 012 t 1.6 1.7t 3.2 3.3

Table 2.2: The effect of the wake drag on the deviation of the rotation speed of the rotor fromideal rotation for turbine flow meter 1 and 2, wheret is the blade thickness.

and it includes the pressure drag due to the wake behind the blades discussed in thepreceding section. To approximate the friction forces on the rotor blades and thehub, boundary layer theory has been used, neglecting centrifugal forces as well asthe radial velocity. In recent years numerical studies on turbine flow meters (VonLavante et al., 2003) show that the flow in the rotor has a complicated 3-dimensionalstructure invoking secondary flows. It should be realised that the theory presentedhere is a very simplified approximation of reality.

In the following sections the effect of these forces on the deviation from idealrotation will be investigated and discussed separately for both meters discussed in2.2.3.

2.5.1 Boundary layer on rotor blades

Boundary layers are formed on the rotor blades as a result of friction. The boundarylayer thickness can be calculated using boundary layer theory and is included in themomentum equation (equation 2.2). We assume that the cascade of rotor bladescan be described as row of rectangular channels with boundary layersat the top andbottom of each channel. We neglect centrifugal forces and assume thatthere is noradial velocity component. The rotor consists ofn rectangular channels with a lengthof Lblade (the length of the blade) and a width ofhblade (the height of the blade). Thedistance between two successive blades is2πr

n− t

cos β. We consider two cases: the

case of a laminar boundary layer and the case of a turbulent boundary layer. Thedisplacement thickness,δ1,bl, the momentum thickness,δ2,bl, of the boundary layerformed in this channel is calculated using the Von Karman equation (2.17). For thelaminar case a third order polynomial is used to describe the velocity profile in theboundary layer. For the turbulent case a 1/7th power law approximation is used. Thevelocity between the blades is corrected for the displacement due to the growth ofthe boundary layers in the channel. Using the mass conservation for incompressible


flow, the out-going velocity component in the x-direction,uout,x, becomes:

uout,x =2πr

2πr − n(2δ1,bl+δ1,wake)cos β

uin . (2.24)

Using the definition of the displacement thickness,δ1,bl, and the momentum thick-ness,δ2,bl, (see Appendix B), for the boundary layer thickness at the end of the chan-nel (the trailing edge of the blade), the equation for the driving torque,Td, becomes:

Td = −∫ rtip

rhub

ρu2

in,x(tanβ)r(

1 − n(δ1,wake+2δ1,bl)2πr cos β

)2 ×

(

2πr − n(δ1,wake + δ2,wake + 2(δ1,bl + δ2,bl))

cosβ

)

dr + 2π

∫ rtip

rhub

ρuin,xωr3dr ,

whereδ1,wake is the displacement thickness caused by the wake andδ2,wake is themomentum thickness caused by the wake (section 2.4.2). The rotation speed of therotor can now be found by determining iteratively at which rotational speedthe totaltorque in the equation of motion (2.13) is zero. This is determined numerically withthe secant method, a version of the Newton-Raphson method. In figure 2.18 the effectof the boundary layers on the two different types of turbine flow meters for steadyincompressible flow with uniform inflow velocity and infinitesimally thin blades andwithout other friction forces.

The laminar boundary layer causes the rotor to rotate faster, because thedisplace-ment thickness of the thicker laminar boundary layers. For the range of calculatedReynolds numbers turbulent boundary layers cause less variation in the deviation.

2.5.2 Friction force on the hub

Not only is there a friction force from the boundary layers on the rotor blades, butalso on the hub of the rotor a boundary layer is formed due to the rotation of the rotor.The shape of this boundary layer is complex and we will approximate this boundarylayer as a boundary layer on a long flat plate of a widthw = 2πrhub−nt, whererhub

is the radius of the hub,n is the number of blades andt is the blade thickness. Thevelocity outside the boundary layer will be assumed constant for simplicity reasonsand equal to the relative velocity:

urel =

√

√

√

√

(

U

1 + nt2π cos βhub

)2

+ (ωrhub)2 , (2.25)


103

104

105

106

−2

0

2

4

6

8

10

12

14

16

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

air at 1 bar, turbulentair at 1 bar, laminarnatural gas at 9 bar, turbulentnatural gas at 9 bar, laminar

(a) turbine meter 1

103

104

105

106

−1

0

1

2

3

4

5

6

7

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)


(b) turbine meter 2

Figure 2.18: The deviation from ideal rotation caused by theboundary layers on the bladesfor turbulent and laminar boundary layers, assuming a uniform inlet velocityprofile, versus Reynolds number, Re= ULblade/ν.

whereU is the velocity at the entrance of the turbine meter corrected for the displace-ment due to the boundary layers (see equation 2.18). Again, we assume that there isno radial velocity and secondary flow. To determine the shear stress,τw, caused bythis boundary layer, two limits are considered. The first case is the upper limitfor theshear stress; the boundary layer starts at the entrance of the rotor. The flat plate hasa length of W

cos βhub, whereW is the width of the rotor andβhub is the angle of the

rotor blades with the rotor axis at the hub of the rotor (see figures 2.1 and 2.2). Thesecond case is the lower limit; the boundary layer starts at the front end of the hub


and continues at the rotor. In this case the flat plate model of the flow has a length ofLhub + W

cos βhub. The empirical expression for shear stress for a turbulent boundaryin

a circular pipe, equation B.20, has been found to be a good approximation for the flatplate (Schlichting, 1979). Using this equation and the equation for boundary layerthickness for turbulent flow for a flat plate, the shear stress,τw, becomes:

τw = 0.0288 ρu9

5

relν1

5x−1

5 . (2.26)

The upper limit of the friction torque relative to the rotor axis due to the boundarylayers on the hub of the rotor,Tfr,hub = Ffr,hubrhub sinβhub, is:

Tfr,hub <wrhub sinβhub

∫ Wcos βhub

0τwdx

=0.036 ρν1

5u9

5

relwrhub sinβhub

(

W

cosβhub

)4

5

,

(2.27)

and the lower limit is;

Tfr,hub >wrhub sinβhub

∫ Lhub+W

cos βhub

Lhub

τwdx

=0.036 ρν1

5u9

5

relwrhub sinβhub

[

(

Lhub +W

cosβhub

)4

5

− L4

5

hub

]

.

(2.28)

Using this in the equation of motion for the rotor (equation 2.13) and assumingsteady uniform flow and infinitesimally thin blades, while neglecting all other frictionforces, including the boundary layer on the rotor blades, the deviation from the idealrotation speed is computed. The result is plotted for different Reynolds numbers infigure 2.19.

As expected the friction force on the hub slows down the rotor. For turbinemeter2, the larger flow meter, the effect is relatively small (at most 0.2%), while for turbinemeter 1, the effect can reach 1.5%. The ratio between the effect of the upper (equation2.27) and the lower (equation 2.28) limits is about 1.5.

2.5.3 Tip clearance

The tip of the rotor blades moves close to the pipe wall of the meter body. Thisimposes an additional drag force on the rotor. The force caused by the flow around thetip is complicated and depends on the size of the clearance and the Reynolds number,but also on the shape and length of the blade tip. In some meters, for example turbinemeter 1, the tip is enclosed in a slot (see figure 2.20).


103

104

105

106

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

air at 1 bar, lower limitair at 1 bar, upper limitnatural gas at 9 bar, lower limitnatural gas at 9 bar, upper limit

(a) turbine meter 1

103

104

105

106

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

ReL

blade

E=

(ω

− ω

id)

/ ωid

* 1

00 (

%)

air at 1 bar, lower limitair at 1 bar, upper limitnatural gas at 9 bar, lower limitnatural gas at 9 bar, upper limit

(b) turbine meter 2

Figure 2.19: The upper an lower limits of the deviations fromideal flow caused by the bound-ary layers on the hub, assuming a uniform inlet velocity profile, as a function ofthe Reynolds number, Re= ULblade/ν.

Thompson and Grey (1970) suggested that the tip clearance drag can beconsid-ered to be similar to the drag in a journal bearing. This results in friction torquecaused by the tip clearance,Ttc of

Ttc =0.078

2 Re0.43tip

ρu2rel,rtip

rtipLbladetn . (2.29)

Here the Reynolds number is defined as Retip = urel,rtip(rout − rtip)/ν, with rout

the radius of the pipe wall of the turbine flow meter.


r o t o r b l a d e r o t o r b l a d e

p i p e p i p e

( a ) ( b )

Figure 2.20: (a) shows an enclosed blade tip, (b) shows bladetip not enclosed

However, if vortex shedding occurs at the front edge of the tip and the bladelength is relatively small, the flow is no longer comparable to the flow in a journalbearing. This makes it very difficult to give a reasonable prediction of thetip clear-ance drag. Therefore, no theory for the tip clearance is incorporatedin the presentcalculations. However, to understand the effect caused by tip friction calibration datawas obtained by Elster-Instromet for three turbine meters with identical geometriesexcept for the height of the tip clearance. These Reynolds curves andthe calculatedReynolds number dependence of the deviation are discussed in section 2.6.3.

2.5.4 Mechanical friction

Turbine flow meters are designed to minimize the mechanical friction. However,mechanical friction will always be present. The main part of the mechanicalfrictionis caused by the bearing friction of the rotor. For a small meter the magnetic pickup and the index counter can cause some additional friction, for larger meters thispart can be neglected. The mechanical friction torque is assumed to be constant,independent of the rotation speed,ω. Experiments were done for turbine meter 1 todetermine the mechanical friction. Two different approaches were usedto determinethe mechanical friction torque: dynamic and static friction measurements. Thesemeasurements are discussed in more detail in section 4.2.3. The friction torquefoundin the experiments are of the same order of magnitude as the data provided by themanufacturer. For turbine meter 1 a mechanical friction torque ofTmech = 5.6×10−6

Nm is assumed. For turbine meter 2 a torque ofTmech = 5.5×10−5 Nm is assumedbased on the data of Elster-Instromet. Again, the equation of motion for the rotor wassolved for steady uniform flow with a rotor with infinitesimally thin blades and noother friction forces than the mechanical friction torque. The deviation is shown infigure 2.21.

The deviation caused by the mechanical friction does not depend on Reynoldsnumber, but depends on the volume flow and density, causing different behaviour at a


103

104

105

106

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

air at 1 barnatural gas at 9 bar

(a) turbine meter 1

103

104

105

106

−14

−12

−10

−8

−6

−4

−2

0

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

air at 1 barnatural gas at 9 bar

(b) turbine meter 2

Figure 2.21: The deviations from ideal rotation caused by the mechanical friction assuminga uniform inlet velocity profile as a function of the Reynoldsnumber, Re=ULblade/ν.

certain Reynolds number for air at 1bar and natural gas at 9 bar. We see from figure2.21 that the mechanical friction is only important at low flow velocities.

2.6 Prediction of the Reynolds number dependence in steady flow

To evaluate the model described above, the results of the model including allfrictionforces discussed in the previous sections and assuming that the flow entering the ro-tor is non-uniform, is compared to the calibration measurements of the two turbine

2.6. Prediction of the Reynolds number dependence in steadyflow 39

meters; turbine meter 1 (Instromet SM-RI-X G250) and turbine meter 2 (more infor-mation about the meters can be found in table 2.1). The measured data is obtainedfrom the manufacturer of the turbine meter Elster-Instromet. The results of the calcu-lations shown in the figures below are found assuming an inlet velocity profilebasedon boundary layer theory with turbulent boundary layers. For the friction on the hubthe upper limit described in section 2.5.2 is used, while tip friction is ignored. Forthe drag caused by either laminar or turbulent boundary layers the predictions for therotor blades are shown in the figures presented below. In the figures presented belowthe prediction for both laminar as well as for turbulent boundary layers onthe rotorblade are shown.

2.6.1 Turbine meter 1

Typical calibration data is obtained for air at 1bar (atmospheric pressure) and naturalgas at 9bar for turbine meter 1 (see figure 2.22).

The measured data of this turbine meter is compared to the calculated deviation,E, for different Reynolds numbers, ReLblade

, using the properties of air at 1barand of natural gas at 9bar and the dimensions given in table 2.1. Because it is notknown whether the boundary layers on the blades are either turbulent orlaminar,both situations are plotted in figure 2.22. The effect of the wake is calculatedfor apressure drag coefficient ofC ′

D = 0.5, based on the experiments of Hoerner (1965) asproposed in section 2.4, for a wake of thicknesswwake = 1

2 t (this leads to a pressuredrag coefficient ofC ′

D = 0.25) and for no wake (C ′D = 0).

The pressure drag coefficientC ′D = 0.5 predicts measurement errors,E, almost 4

times lower than the measured data. To be able to obtain data similar to the measureddata an unrealistic high drag coefficient would be needed. It is unrealisticthat themodel shows better agreement with the measured data using a drag coefficient muchhigher than that of the blunt object measured by Hoerner (1965). Otherreasons forthe observed high values ofE have to be found. Deviation between theory and exper-iment could for example be due to the blade experiencing a nonzero lift coefficient ata zero angle of attack (see section 2.4.1 and figure 2.17). This additional lift wouldresult in a higher rotor speed and accordingly a higher error,E. Another effect thatcan account for the deviation between theory and experiment is the inflow velocityprofile. The measured inlet velocity profiles described in section 2.3.2 are used tocalculate the deviation between actual rotation speed and ideal rotation speed. Infigure 2.23 the deviation from ideal flow is plotted using the measured inlet velocityprofile in the calculations. The calculations have been carried out for bothturbu-lent and laminar boundary layers on the rotor blades and a pressure drag coefficientC ′

D = 0.5. The actual velocity profile in the turbine meter at the inlet of the rotorinduces a larger rotation speed, however, it cannot explain the difference between the


103

104

105

106

−10

−5

0

5

10

15

20

25

30

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)


C’D=0.5

measurements

C’D=0.25

C’D=0

Figure 2.22: The deviations from rotation versus Reynolds number, Re= ULblade/ν forturbine meter 1 in air at 1bar and natural gas at 9bar. The measured datafor air at 1 bar are indicated by circles• , the solid symbols represents thedata for a meter with standard blades, the open circles are for a blade witha chamfered leading and trailing edge. The data indicated bytrianglesH arefor natural gas at 9bar. The lines represent the calculated data, the results ofthe calculations are for turbulent boundary layers as well as laminar boundarylayers on the rotor blades at three different pressure drag coefficients (solid,dashed and dotted lines).

theory and calibration data by itself.The standard rotor blade has a rounded leading edge and a chamfered trailing as

shown in figure 2.24(a). For a calibration measurement with air at 1bar the standardrotor was replaced by a rotor with blades, where the leading and the trailing edge areboth chamfered (figure 2.24(b)). These results are plotted in figure 2.22as the opendots. By changing the shapes of the blades, the shape of the error curve as functionof Reynolds changes. In the present case the effect of the shape ofthe leading edgeof the rotor blade is quite small.

To compare the shape of the measured curve with the shape predicted by themodel, we shifted the calculated data by 12.7% (see figure 2.25). The small devi-ations caused by the different blade profile cannot be explained using this globalmodel. The measured data resemble the prediction of the model with turbulentboundary layers on the rotor blades much more than the model with laminar boundary


103

104

105

106

−10

−5

0

5

10

15

20

25

30

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)


C’D=0.5

Figure 2.23: The deviations from ideal rotation versus Reynolds number, Re= ULblade/ν forturbine meter 1 in air at 1bar and natural gas at 9bar. The symbols indicatethe results of the calculations using the measured velocityprofile for laminarboundary layers (�) and turbulent boundary layers (♦) on the rotor blades forC ′

D = 0.5. The lines represent the calculated data using a turbulent boundarylayer model for the inlet velocity profile. The calculationsare performed for tur-bulent boundary layers as well as laminar boundary layers onthe rotor bladesat C ′

D = 0.5 (solid, dashed, dotted and dashed-dotted lines). The measureddata for air at 1bar are indicated by circles•. The data indicated by trianglesH are for natural gas at 9bar.

(a) round leading edge (b) chamfered leading edge

Figure 2.24: A schematic drawing of the rotor (a) with rounded leading edge and (b) with achamfered leading edge used in the measurements


103

104

105

106

4

6

8

10

12

14

16

18

20

22

24

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

Figure 2.25: The deviations from ideal rotation versus Reynolds number, Re= ULblade/νfor turbine meter 1 in air at 1bar and natural gas at 9bar. The calculated datais shifted upwards by 12.7%. The measured data for air at 1bar are indicatedby circles• , the solid symbols represents the data for a meter with standardblades, the open circles are for rotor blades with a chamfered leading edge. Thedata indicated by a triangleH have been measured in natural gas at 9bar. Thelines represent the calculated data, the calculations are performed for turbulentboundary layers (dotted: air at 1bar; dash-dot: gas at 9bar) as well as laminarboundary layers (solid: air at 1bar; dashed: gas at 9bar) on the rotor blades.

layers over the entire range of Reynolds numbers considered. For the inlet velocityprofile a turbulent boundary model was used. Figure 2.23 shows that themeasuredvelocity profile changes the shape of this curve considerably. We conclude that themanufacturer has used modifications of blade tip geometry (figure 2.20) in order tocompensate for this Reynolds number dependence of the main flow velocity profile.

2.6.2 Turbine meter 2

The calibration measurements of turbine meter 2 were compared to the results ofthe calculations for the model assuming either turbulent or laminar boundary layerson the blade. Again, this is done at different pressure drag coefficients,C ′

D = 0.5,0.25 and 0. The results are shown in figure 2.26. Again, a drag pressure coefficient


103

104

105

106

−10

−5

0

5

10

15

20

25

30

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)


measurementsC’

D=0.5

C’D=0

C’D=0.25

Figure 2.26: The deviations from ideal rotation versus Reynolds number, Re= ULblade/νfor turbine meter 2 in air at 1bar and natural gas at 9bar. The measureddata for air at 1 bar are indicated by circles• and for natural gas at 9barby trianglesH. The lines represent the calculated data, the calculationsarecarried out for turbulent boundary layers as well as laminarboundary layerson the rotor blades at three different drag coefficients (solid, dashed and dottedlines).

of C ′D = 0.5 results in deviations from ideal rotation considerably lower than the

measured data, as was the case for turbine meter 1. The deviation between the theoryand the experiments could be caused by the nonzero lift coefficient at a zero angle ofattack (figure 2.17). This additional lift can cause a higher error,E. A comparisonbetween the shape of the measured curve and the calculated curve is made by adding6.5% over the whole Reynolds range (see figure 2.27). The deviation measured forturbine meter 2 is in general lower than the deviation obtained for turbine meter 1.It is possible that it is the influence of not having full fluid guidance at the tipofthe rotor. As explained in section 2.2.3 a deviation from full fluid guidance reducesthe rotation speed and the error,E. The shape of the curve predicted by the modelagrees very well with the measured data assuming laminar boundary layers for theair set up (ReLblade

< 105) and turbulent boundary layers for the natural gas set up(ReLblade

> 104). This could explain the gap between the two different measurementconditions at the same Reynolds numbers. Since the measurement have beendonefor two different set ups it is possible that the different conditions in the set up can


103

104

105

106

−2

0

2

4

6

8

10

12

14

16

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

Figure 2.27: The deviations from ideal rotation versus Reynolds number, Re= ULblade/νfor turbine meter 1 in air at 1bar and natural gas at 9bar. The calculateddata is shifted upwards by adding 6.5%. The measured data forair at 1 barare indicated by means of circles•. The data indicated by means of trianglesH are for natural gas at 9bar. The line represents the calculated data, thecalculations are carried out for turbulent boundary layers(dotted: air at 1bar;dash-dot: gas at 9bar) as well as laminar boundary layers (solid: air at 1bar;dashed: gas at 9bar) on the rotor blades.

force the boundary layers to be either turbulent or laminar. No measurements weredone to ascertain the velocity profile at the inlet of the rotor for turbine meter 2. Wecannot determine more accurately the influence of the inflow velocity profile inthiscase. However, we expect this would influence the deviation,E.

Comparison between turbine meter 1 with ”enclosed” blad tips and turbine meter2 with normal blade tip clearance (figure 2.20) clearly illustrates the importanceof thetip clearance on the response of the flow meter. The Reynolds number dependenceof the measurement error,E, is much stronger for turbine meter 2 than for turbinemeter 1.


diameter,D (m) 0.280blade thickness,t (mm) 3.0

number of blades,n 24rhub/D 0.359rout/D 0.500S/D 2.707W/D 0.136

hblade/D = (rtip − rhub)/D 0.138, 0.137, 0.136Lhub/D 1.532

Table 2.3: Dimensions of the turbine flow meters 3, 4 and 5 withdifferent height of the tipclearance, whererhub is the radius of the hub,rout is the radius of the outer wall,rtip is the radius at the tip of the blades,S is the pitch,W is the width of the rotor,hblade is the height of the blade andLhub is the length of the hub in front of therotor. Except for the blade thicknesst and the number of bladesn, all values aremade dimensionless with the diameter,D.

2.6.3 Effect of tip clearance

As mentioned in section 2.5.3 some extra attention is given to the effect of tip clear-ance. Because no satisfactory model has been found to describe the friction causedby the flow through the gap, this effect is not included in our model. To estimatethe effect of different tip clearance, our model is compared to a series of turbine me-ters with the same geometry, but with different height of the tip clearance,htc =rout − rtip = 1, 2, 4mm (see figure 2.28). As for turbine meter 2 the tip is not en-closed in a wall cavity (figure 2.20). The dimensions of the meter are given intabel2.3.

r o t o r b l a d e

p i p eh t c

r o u t r t i p

Figure 2.28: Tip clearance of a blade that is not enclosed.


The calibration data for all three turbine meters (turbine meter 3, 4 and 5) areplotted in figure 2.29. The measurements show a decrease inE for increasing tipclearance height,htc.

The theoretical results are shown in figure 2.30, assuming an inlet velocity profilebased on turbulent boundary layers, the drag coefficient is taken to beC ′

D = 0.5and using the upper limit of the hub friction. Although there is no model for tipclearance drag, there is still an effect, because the height of the blade decreases forincreasing gap height,htc. For larger blade heights less of the blade is rotating in theboundary layer at the outer wall of the pipe, this causes the increase of rotation speedof the rotor for larger tip clearance as can be seen in figure 2.30. If the effect of tipclearance drag suggested by Thompson and Grey (1970), equation 2.29, is taken intoaccount, the results do not change significantly. Because the mechanicalfriction ofthis particular turbine meter has not been measured, it was assumed to be equal to thatof turbine 2. It is however clear from the figures that in reality the mechanical frictionis larger than the assumed mechanical friction, therefore the calculated deviation,E,decreases considerably slower for small Reynold numbers than in the measured data.

It is clear that the theoretical model described in this chapter is not able to de-scribe the effect of tip clearance drag correctly. The results of the experiments shownin figure 2.29 display exactly the opposite behaviour of the results obtained with ourmodel shown in figure 2.30. As the tip gap increases our model predicts an increasingrotational speed, while the experiments indicate a reduction of rotational speed.

2.7 Conclusions

To predict the influence of modifications of the geometry of a turbine meter andofchanges in fluid properties, a theoretical model of the behaviour of the turbine flowmeter in steady flow is necessary. The flow in a turbine meter is 3-dimensional andvery complicated. We consider a very simplified model. The model presented inthischapter makes it possible to calculate for a chosen geometry the Reynolds numberdependence of the deviation between the rotor response and that of an ideal rotor.

We found that the shape of the velocity profile is important. More accurate mea-surements should be carried out to provide a better understanding of this effect.

In the proposed model we included the effect of the friction due to skin frictionand a wake displacement effect. The wake thickness can be tuned to match the dragcoefficient measured for two-dimensional models of rotor blades in a wind tunnel.However, a comparison with measured data shows that it is not possible to accountfor the high rotation speed of the rotor compared to ideal rotation by modifyingthiswake thickness only. A possible explanation is that the flow also generates alift forceon the rotor blades at zero angle of attack (figure 2.17). The lift causesthe rotor torotate faster. A lift force on a single rotor was measured in wind tunnel experiments.

2.7. Conclusions 47

103

104

105

106

107

0

2

4

6

8

10

12

14

16

18

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)

tc=1 mm; air at 1 bartc=1 mm; natural gas at 9 bartc=2 mm; air at 1 bartc=2 mm; natural gas at 9 bartc=4 mm; air at 1 bartc=4 mm; natural gas at 9 bar

Figure 2.29: The measurement deviation from ideal rotationfor turbine meters with tip clear-ance height of 1, 2 and 4mm versus Reynolds number.

103

104

105

106

107

0

2

4

6

8

10

12

14

16

18

ReL

blade

E =

(ω

− ω

id)

/ ωid

* 1

00 (

%)


Figure 2.30: Calculated deviation of ideal rotation for turbine meters with tip clearanceheight,htc = 1 mm (solid line),htc = 2 mm (dashed line) andhtc = 4 mm(dotted line). Calculation are for laminar as well as turbulent boundary layerson the rotor blades and the solid symbols represent the data using the equationof tip clearance drag suggested by Thompson and Grey (1970),equation 2.29.The drag coefficient used isC ′

D = 0.5.


Measurements on a cascade are needed to determine this lift force for a turbine metertype of rotor.

The present model gives an adequate prediction of the shape of the Reynoldsnumber dependency of the rotor response. However, the effect of changing the shapeof the rotor blades and the effect of the tip clearance cannot be explained.

Experiments clearly demonstrate that the flow around the blade tip has a stronginfluence on the Reynolds number dependence of the response of the rotor.

3

Response of the turbine flow meter onpulsations with main flow

3.1 Introduction

Gas turbine flow meters can reach high accuracy, generally of the orderof 0.2%. Thisaccuracy can only be attained for optimal flow conditions. Acoustic perturbations caninduce significant systematic errors. A theoretical prediction of the errorwould allowa correction in the volume flow measurement.

In recent years research has been carried out to determine the pulsation errorduring the measurement and correct for this error instantaneously. Atkinson (1992)developed a software tool to solve the equation of motion of the rotor (equation 3.5)and used the magnetic pickup registering the passing of a rotor blade to calculate thereal volume flow. This method can only be used if the amplitude of the pulsations canstill be detected in the turbine signal. As the amplitude of the pulsations in the tur-bine meter signal decreases rapidly with increasing frequency, it is difficult to predictthe real flow for high frequency pulsations. Another tool was developed describedby Cheesewright et al. (1996), called the ’Watchdog System’. This system also usesthe equation of motion of the rotor (equation 3.5), but now an accelerometer isusedto measure the flow noise of a valve or bend. Watchdog is designed for pulsationfrequencies less than 2Hz. We actually focus on the behaviour of the rotor at highfrequencies for which the rotor inertia has integrated fluctuations in rotor speed. Theerrors we consider are due to non-linearities.

Assuming quasi-steady, incompressible flow and neglecting friction forcesa re-lationship can be found between the velocity pulsations and the measurement error(Dijstelbergen, 1966). Experiments have been carried out in the past byLee et al.

50 3. Response of the turbine flow meter on pulsations with main flow

(2004), Jungowski and Weiss (1996), Cheesewright et al. (1996)and McKee (1992).These experiments indicated that this basic theory can be used for low frequency pul-sations. To explore the limits of the validity of this theory, a set up was build at theEindhoven University of Technology.

In our experiments care was taken to determine accurately the amplitude of thevelocity fluctuations at the rotor. This was found to be a limitation in the experimentsreported in literature. With the more accurate determination of the acoustic fieldit is possible to detect small deviations from the basic theory. With this set up itwas possible to measure the influence on the flow meter response of the acousticperturbation with velocity amplitudes from 2% of the main flow velocity up to twicethe main flow velocity for frequencies from a fewHz up to 730Hz. In this chapterthe basic theory is discussed, after this the set up is described and the experimentalprocedures are discussed. We then show the results of the measurementsand discussthese results.

3.2 Theoretical modelling

3.2.1 A basic quasi-steady model: A 2-dimensional quasi-steady model for arotor with infinitesimally thin blades in incompressible flow

If the rotor is modelled as a 2-dimensional cascade of infinitesimally thin blades inanincompressible, frictionless, steady flow, the integral mass and momentum equationapplied to a fixed control surfaceCS with an outer normaln (equations 2.1 and 2.2)reduce to

ρ0

∫∫

CS

v · ndS = 0 , (3.1)

ρ0

∫∫

CS

v (v · n) dS = F bf , (3.2)

whereρ0 is the fluid density,v is the velocity vector andF bf are the forces imposedon the fluid by the turbine blades. The control volume,CS, is chosen as shown infigure 3.1 and we assume that there is no pressure drop over the cascade. Becauseinfinitesimally thin blades and frictionless flow are assumed, the surface areaof in-flow is equal to the surface area of outflow. It follows from equation 3.1 that the axialcomponent of the incoming velocity,uin, is equal to the outgoing velocity,uout,x;uin = uout,x. It is assumed that the flow enters the rotor without any angular mo-mentum and that the flow leaves the rotor with a velocity aligned with the blades (seefig. 3.1). This is a realistic assumption if the chord length of the blades,Lblade, islarge compared to the distances between the blades,s, i.e. for cascades the ratio,

3.2. Theoretical modelling 51

w ru i n

u o u t , y

w r

u o u t , x = u i n

b

u r e l

b

C S

xy

n

n

t

L b l a d e

u r e l

W

Figure 3.1: Two-dimensional representation of the flow entering and leaving the rotor mod-elled as a cascade

s/Lblade, should be smaller than0.7 (Weinig, 1964). By considering the flow in areference frame attached to the rotor, this implies that :

tanβ =ωr + uout,y

uout,x. (3.3)

We assume further that there is no swirl in the incoming flow, so thatuin,y = 0 andthat the inflow is uniform. The momentum equation in the y-direction becomes

ρ0Auin (uin tanβav − ωR) = Fbf,y , (3.4)

whereω is the angular rotation velocity of the rotor,A is the cross-sectional area ofthe rotor,βav is the average blade angle andR is the root-mean-square radius of the

inner and outer radius of the meter,rin androut respectively, i.e.R =

√

(r2in+r2

out)2 .


The force exerted on the fluid by the blade,Fbf,y, is equal and opposite to the forceexerted by the fluid on the blade,Ffb,y. The fluid induces a torque on the rotor,Tfb = Ffb,yR, accelerating the rotor. Using the equation of motion of the rotor, weget:

Irotordωdt

= ρ0Auin(uin tanβav − ωR)R− Tf , (3.5)

whereIrotor is the moment of inertia of the rotor andTf is torque on the rotor causedby the friction forces.

We assume periodic pulsationsu′in around an average velocityuin so thatuin =

uin + u′in. We neglect the friction torque, Tf

ρu2inR3 ≪ 1. We assume that the rotation

of the rotor is constant in spite of the pulsations. The integration time of the rotoris much longer than the period of the imposed acoustic pulsations. If the torqueisaveraged over one period, equation 3.5 reduces to:

1

T

∫ T

0ρ0(uin + u′in)A

[

(uin + u′in) tanβav − ωR]

Rdt =

ρ0AR[

(u2in + u′2in) tanβav − uinωR

]

= 0 , (3.6)

with T is the period of the pulsations. From this equation the angular rotation veloc-ity, ω, caused by pulsating flow is obtained

ω =uin tanβav

R

(

1 +u′2inu2

in

)

. (3.7)

Using equation 1.3, i.e.ωid = uin tan βav

Rfor the ideal angular rotation velocity

without pulsation, the error caused by periodic pulsations becomes:

(Epuls)id=ω − ωid

ωid

=

(

u′2inu2

in

)

. (3.8)

This means that sinusoidal pulsations,uin = uin + |u′in| sin(ωt), induce a systematicerror of

(Epuls)id=

1

2

( |u′in|uin

)2

. (3.9)

3.2.2 Practical definition of pulsation error

In the previous section we defined a deviation(Epuls)idbetween angular velocity,ω,

for steady rotation and the ideal angular rotation velocity,ωid, in absence of pulsa-tions:

(Epuls)id=ω − ωid

ωid, (3.10)

3.3. Experimental set up 53

whereωid = uin tan βav

R. In experiments we use the steady angular velocityω0 as

reference instead ofωid, hence:

Epuls =ω − ω0

ω0. (3.11)

In order to illustrate the difference between this ideal pulsation error,(Epuls)id, and

the definition of the pulsation error used in the experiments,Epuls, we consider theinfluence of a constant mechanical friction torque,Tmech on an ideal rotor. Usingequation 3.6 we find in absence of pulsations:

ω0 = ωid −Tmech

ρ0AR2uin, (3.12)

while due to pulsations we would reach a steady rotation of angular velocity:

ω =uin tanβ

R

(

1 +u′2inu2

in

)

− Tmech

ρ0AR2uin. (3.13)

Hence, we would predict a pulsation error,(Epuls)exp:

(Epuls)exp=

(

ωid

ω0

)

u′2inu2

in

, (3.14)

corresponding to(Epuls)id=

u′2in

u2in

multiplied by a factorωid

ω0.

Other reasons for a deviation between the ideal pulsation error,(Epuls)id, and the

measured pulsation error,Epuls defined by equation 3.11, is the unsteadiness of theflow at high Strouhal numbers. Our aim is to provide quantitative information aboutthis Strouhal number dependence.

3.3 Experimental set up

A dedicated set up has been built at Eindhoven University of Technology to study theinfluence of pulsations on gas turbine meters. A high pressure reservoirwith dry air at60bar (dew point -40◦C) is connected to a test pipe of 0.10m diameter, and a lengthof 3.2m. At the open end of this pipe a turbine flow meter (Instromet type SM-RI-XG250) is placed. The flow through the turbine flow meter is controlled by meansof a valve placed at the upstream end of the test pipe. By adjusting this valve, thecritical pressure at the valve is reached, resulting in a velocity,u∗, at the valve equalto the local speed of sound,c∗. We have a Mach number of unity,M = u∗/c∗ = 1and a so-called ”choked” flow. This provides a constant mass flow, independent of


p 1 p 2 p 3 p 4 p 5 p 6p 7 p 8

h w 1

t e m p e r a t u r e

h w 2

h o t w i r e m e a s u r e m e n t

s i r e n

a i r -r e s e r v o i r c h o k e d

v a l v e

s i g n a lg e n e r a t o r

p u l s es h a p e r

A

CB

D

E

F

G

t r i g g e r( 8 )

( 4 )8 xS & H+

f i l t e r

Figure 3.2: Experimental set up: A high pressure reservoir of dry air (A) is connected witha pipe (B) to a turbine meter (D). The flow is being controlled by an adjustablevalve (C) creating choked flow with constant mass flow. Pulsations can be in-duced by a loudspeaker (E) or a siren (F). The pulsations are measured withsix pressure transducers (p1,p2,p3,p4,p5 and p6) along in the pipe (B) and twopressure transducers (p7 and p8) placed within the turbine meter (D). Velocitypulsations can be measured with two hot wires (hw1 and hw2) placed within theturbine meter (D). The rotation of the rotor of the turbine meter is being measuredby a probe detecting the passing of a rotor blade (G).

perturbations in the flow downstream of the valve. The conditions of the reservoir,p0 andT0, and the valve opening determine the mass flow.

Pulsations in the test pipe downstream of the choked valve can be induced byusing a loudspeaker placed at the downstream open end of the set up orby meansof a siren placed downstream of the valve. The loudspeaker (SP-250P) is controlledusing a signal generator (Yokogawa FG120) driving a power amplifier (AIM WPA301A). The siren is described by Peters (1993). The siren has a frequency range froma 10Hz up to 1000Hz. A bypass allows variations in the ratio,uac/u0, of acousticvelocity, uac, and the main flow velocity,u0. The siren is a much more efficientsound source than the loudspeaker, by tuning it to the resonance frequencies of theset up, the ratio of acoustic to main flow velocities,uac/u0, can reach values up to2. Between the siren and the valve a volume is placed, a pipe with a length 1.22mand a internal diameter of0.21 m. Except for the core of the pipe with a diameter of0.05m, this pipe is filled with porous material (Achiobouw acoustic foam D80). Toavoid chocking at the siren, the opening of the bypass was increased, while the siren

3.3. Experimental set up 55

pressure transducer in pipe distance

1 -1.566m2 -1.212m3 -0.960m4 -0.400m5 -0.265m6 -0.205m

pressure transducer in turbine meter distance

7 0.07775m8 0.1075m

Table 3.1: Position of the pressure transducer placed in theset up. The distances are mea-sured from the upstream end of the turbine meter, where the positive direction isthe flow direction.

was turned off, up to the point at which changes in volume flow could no longer beobserved. Only measurements using a significantly larger opening of the bypass thanthis critical point are performed. Goog agreement between the measurements usingthe loudspeaker and using the loudspeaker confirm that there was no chocking.

The acoustic pressure in the set up is measured by means of eight piezo-electricgauges placed flush at the pipe wall. Six pressure transducers (three Kistler type 7031and three PCB type 116A) are placed in the pipe upstream of the turbine metereach atrandomly chosen distances (see table 3.1). Two other pressure transducers (PCB type116A) are placed within the turbine meter, 0.010m and 0.040m upstream of the ro-tor of the turbine meter (see figure 3.3). The signals from the pressure transducers areamplified using charge amplifiers (Kistler type 5011). They are acquired bymeans ofa PC using an 8 channel Sample and Hold module (National Instrument SCXI 1180)and a DAC card (PCI MIO-16E-I) controlled by LabView software. Thepressuretransducer and charge amplifier combinations are calibrated in a differentset up. Inthis calibration set up the transducer is placed next to a reference microphone flush ina closed end wall of a 1.0m long pipe (diameter 0.07m). Plane waves are generatedby a loudspeaker placed at the opposite end of the pipe. All pressure transducers arecalibrated against the reference pressure transducer for frequencies between 24Hzand 730Hz, i.e. the frequencies used in our experiments. The acoustic velocity ofthe pulsations can also be measured using two hot wire anemometers (Dantec type55P11 wire diameter 5µm with 55H20 support) placed 0.010m upstream of the tur-bine meter (see figure 3.3). Accurate measurements of the amplitude of the velocitypulsations are only possible if the ratio between the acoustic velocity amplitude and


p r e s s u r e t r a n s d u c e rh o t w i r e a n e m o m e t e r s

f l o w s t r a i g h t n e r r o t o r

(a)

p r e s s u r e t r a n s d u c e r s

h o t w i r e a n e m o m e t e r

(b)

Figure 3.3: The placement of the pressure transducers in theturbine meter. (a) shows aschematic, simplified drawing of a cross-section of the turbine flow meter and (b)shows a photograph of the turbine meter. One pressure transducer and two hotwires are placed at the same distance upstream from the rotor(1 cm), equallydistributed around the perimeter of the meter.

the main flow velocity is small enough to avoid flow reversal,uac/u0 < 1. The hotwire makes no distinction between forward and reversed flow. The signalsof the hotwire anemometers are processed with a constant-temperature anemometer module(Streamline 90n10) in combination with dedicated Dantec application software. Theanemometer can follow velocity fluctuations up to 50kHz. The signals are recordedon a PC in the same way as the signals of the pressure transducers. The hot wireanemometers are calibrated against a Betz water micromanometer (± 1Pa) in a sep-arate free-jet set up in the velocity range 2m/s to 40m/s. The output is fitted usinga power law description. This results in accuracies of about 1% for velocities above8m/s and 5% for velocities from 2 to 8m/s.

The time-averaged volume flow can be measured using the turbine flow meter (In-stromet type SM-RI-X G250), using the calibration data provided by Elster-Instrometfor normal flow conditions. In the absence of pulsations the volume flow measuredhas an accuracy of 0.2 % in the range of6 × 10−3 m3/s to 1 × 10−1 m3/s. Therotation of the rotor of the turbine meter is detected by means of a so-called ”reproxprobe”, a magnetic pickup generating an inductive pulse when a rotor blade passesthe probe. These pulses are converted to electronic pulses and then modified intoproper TTL pulses by means of the signal generator. The time interval between theTTL pulses is registered using a counter board (PCI 6250 NI), insertedin a PC, withan accuracy of 50ns. These intervals are converted to the rotation period of the ro-tor by multiplying by the number of rotor blades (n=16). Due to small differencesin blade geometry the measured rotor speed is not constant during a rotation. An

3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 57

average rotor speed is calculated for each rotation.

3.4 Determination of the amplitude of the velocity pulsations at thelocation of the rotor

When the loudspeaker or the siren is turned on, velocity pulsations are generated. Thevelocity in the set up can be described by the average main flow,uin, and a periodicfluctuating part,u′in. To investigate the effect of the acoustic perturbations on the flowmeasurements of the turbine meter, it is necessary to determine the velocity pulsationsat the position of the rotor. It is impossible to measure the velocity pulsations exactlyat the rotor. The measured data has to be extrapolated to the rotor position. Byusing the measured pressure fluctuations obtained by the microphones, theacousticvelocity at the rotor is determined by using an acoustic model.

3.4.1 Acoustic model

p 1 +

p 1 -

p 2 +

p 2 - p 3 -

p 3 +

t u r b i n e f l o w m e t e rp i p e

Figure 3.4: A schematic illustration of the acoustic model used to determine the amplitude ofthe velocity pulsations at the position of the rotor.

For the acoustic model the test pipe including the turbine meter is divided intothree parts with different cross-sectional areas (see figure 3.4). The first part is thepipe leading to the turbine meter with a diameterD = 0.10m. The pipe has a cross-sectional area of8.4 × 10−3 m2. As can be seen in figure 3.3 the core of the turbinemeter has a complicated shape around the rotor. In the acoustical model, the turbinemeter will be described as two cylindrical parts changing abruptly in cross-sectionalareas. The first part of the turbine meter is the front part of the flow straightener andhas a length of 0.037m and a cross-sectional area of7.3 10−3 m2. The second part isthe main part of the turbine meter and has a smaller cross-sectional area of3.8 10−3


m2. It is assumed that the acoustic field can be described within each segment asplane waves for frequencies up to the critical frequency of the pipe,fc = c0/(2D) ≈1.7 kHz. Harmonic plane waves are described by the d’Alembert solution of theone-dimensional equation:

p′j(x, t) = pj(x)ei2πft = p+

j ei(2πft−k+

j x) + p−j ei(2πft+k−

j x) , j = 1, 2, 3(3.15)

with pj the complex amplitude andf the radial frequency. In this case of an uniformmain flow, k+

j and k−j are the complex wave numbers of the waves travelling inpositive and negative direction, respectively. The wave numbers are defined as:

k+j =

2πf/c01 +M

+ (1 − i)αd, k−j =2πf/c01 −M

+ (1 − i)αd . (3.16)

The imaginary part of the wave number represents the damping coefficientcaused byviscous-thermal effects. In a quiescent flow in smooth cylindrical pipes,damping ofplane waves by viscous-thermal damping can be described by a damping coefficient,αd (Kirchhoff (1868), Tijdeman (1975), Pierce (1989))

αd =1

rjc0

√

νπf

(

1 +γ − 1√

Pr

)

, (3.17)

whererj is the radius of pipe segmentj, ν is the kinematic viscosity,γ is the Pois-son’s ratio and Pr is the Prandtl number. For air at room temperature and atmosphericpressure the following values are used:ν = 1.5×10−5 m2/s, γ = 1.4 and Pr= 0.72.Although this damping coefficient is deduced for quiescent flow, it provides a goodapproximation of the effect of damping for frequencies such that the acoustical vis-

cous boundary layer,δ =√

νπf

, is thinner than the viscous sublayer10δ+ with

δ+ = ν√

ρτw

(Peters (1993), Ronneberger and Ahrens (1977) and Allam andAbom

(2006)). If we use theδ+ for smooth cylindrical pipes, we find that this is valid forour experiments.

At the abrupt transitions in cross-section the integral formulation of the conser-vation of mass flow,m′, and total enthalpy,B′ are used for compressible potentialflow (Hofmans, 1998);

m′1 = m′

2 ,

B′1 = B′

2 ,

m′j =

Aj

c0

(

p+j e

−ik+

j x(1 +Mj) − p−j eik−

j x(1 −Mj))

,

B′j =

1

ρ0

(

p+j e

−ik+

j x(1 +Mj) + p−j eik−

j x(1 −Mj))

.

(3.18)


By introducing a matrixM and a vector[pm] with the pressures measured atthe microphones, the solution of the system of the equations 3.18 and the equations3.16 at the positions of the microphones can be computed. The system of equationbecomes:

[

p+j

p−j

]

=(

MTM)−1

MT · [pm] (3.19)

From this system of equation the least-square solution of the plane wave amplitudesp+

j andp−j is determined. Using

uac =p+3 e

−ik+

3xr − p−3 e

ik−

3xr

ρ0c0, (3.20)

the velocity pulsations at the position of the rotor,xr, is calculated.

3.4.2 Synchronous detection

To analyse the measured pressure signals synchronous detection, ’lock-in’, is beingused during post-processing. With synchronous detection, it is possibleto measurethe phase and amplitude at a certain reference frequency using a reference signal.The reference signal,sref , has to be a well-defined signal: we will use a sine wave,sref (t) = sin (2πft). When the measurements are carried out by using a loudspeakerto induce pulsations, the signal driving the loudspeaker is used as reference. Whenthe siren is used to induce pulsations, one of the pressure transducers isfiltered outdigitally using a second order band-pass filter to produce a sinusoidal reference sig-nal. From the sine wave reference signal a cosine wave signal is obtained by shiftingthe phase byπ/2. The Hilbert transform routine of Matlab is used to obtain theshifted reference signal. The transducer signals are multiplied by the sine referencesignal and integrated over a integer number of oscillation periods to extractthe am-plitude of thesin(2πft) component of the signal. The same procedure is repeatedfor the cosine reference signal to obtain the amplitude of thecos(2πft) componentof the signal. Integration is done typically over a few hundred periods.

3.4.3 Verification of the acoustic model

To investigate the accuracy of the procedure for the determination of the acousticvelocity several approaches are used. The pressure transducersplaced in the turbinemeter are placed close to the rotor. The rotation of the rotor, the wake of the guidingvanes of the flow straightener and the abrupt transitions in cross-sectioncan cause in-terference on these pressure measurements. This can be investigated byexcluding thetwo microphones placed within the turbine meter. In figure 3.5 an example is given


of an experiment in which pulsations were induced at 164Hz. The figure shows thepressure amplitude, the added upstream and downstream travelling pressure waves.It can be found that the difference between these two models is small, in the orderof a few percent in velocity amplitude, depending on frequency and standing wavepattern.

−2 −1.5 −1 −0.5 0 0.50

100

200

300

400

distance from upstream end of turbine meter (m)

pres

sure

am

plitu

de (

Pa)

positionrotor

(a) all pressure transducers

−2 −1.5 −1 −0.5 0 0.50

100

200

300

400


pres

sure

am

plitu

de (

Pa)

positionrotor

(b) pressure transducers in the turbine meterare excluded

Figure 3.5: The pressure amplitude of the standing wave in the set up during a measurementat a frequency off = 164 Hz with a mainstream velocity in the pipe ofu0 = 2m/s. The dots represent the measured pressure amplitude of the pressure trans-ducers. Figure (a) shows a example of a measurement were all eight pressuretransducers in the set up are used, (b) shows the standing wave predicted whenthe two pressure transducers in the turbine meter are not used. The different linesindicate the three different parts of the acoustic model (figure 3.4).

The accuracy of the acoustic model depends on the position of the pressure nodesof the standing wave. If the pressure node is located around the rotor position, smalldeviations in the pressure wave induce small deviations in the velocity amplitude,because the velocity is rather uniform around a pressure node. When the rotor isclose to a pressure antinode large errors in velocity amplitude can be induced bysmall deviations. Measurements are only considered when the acoustical velocitycan be determined accurately. In figure 3.6 an example is given of a measurementat 362Hz, for which the position of the rotor is close to a pressure maximum. Theresults of this experiment were therefore not used in our analysis.

To illustrate this, the velocity amplitude was calculated at the front of the rotorand at the back of the rotor (the rotor has a width of2 cm). In the case shown in figure3.6(a), where the rotor position is around a pressure antinode, the velocity amplitudechanges over the width of the rotor with 28%, while in the figure 3.6(b) the velocityamplitude changes with 10%. Besides the location of the rotor position in reference tothe standing wave, the frequency also plays an important roll. In figure 3.7examples


−2 −1.5 −1 −0.5 0 0.50

100

200

300

400


pres

sure

am

plitu

de (

Pa)

positionrotor

(a) rotor position around pressure maximum

−2 −1.5 −1 −0.5 0 0.50

100

200

300

400


pres

sure

am

plitu

de (

Pa)

positionrotor

(b) rotor position not in pressure maximum

Figure 3.6: The pressure amplitude of the standing wave in the set up for a measurement at afrequency off = 362 Hz with a mainstream velocity in the pipe ofu0 = 2m/s.The dots represent the measured pressure amplitude of the pressure transducers.Figure (a) shows a example of a measurement where the rotor islocated close toa pressure antinode, (b) shows a measurement at the same frequency and mainstream velocity with the rotor not as close to the pressure maximum. The differentlines indicate the three different parts of the acoustic model (figure 3.4).

of a measurement at24Hz and a measurement at730Hz are shown. If we evaluatethe accuracy as mentioned above, for 24Hz the velocity amplitude changes over therotor with less than 0.1%, while at 730Hz this is about 3 percent. In this example(730Hz) this change in velocity amplitude is still relatively small, because the rotoris positioned at a pressure node.

To verify the velocity amplitude found with the acoustical model further, the ve-locity amplitude is measured 1cm upstream of rotor by means of two hot wiresplaced at different positions. Two hot wires were used to account forthe complicatedflow profile behind the blades of the flow straightener. The local relative velocitypulsations for|u′|/u0 < 1, can be compared with the relative velocity amplitudecalculated with the acoustical model based on the pressure measurements. The mea-surements of the velocity amplitude with the hot wires are within 10% in agreementwith the acoustical model for velocities higher than 2m/s. Below 2m/s the cali-bration of the hot wire is problematic. We will discuss the hot wire measurementsfurther in section 3.4.4.

The siren generated block pulses in volume flow, which drives many harmonicsof the fundamental frequency. But by using the siren only at resonance frequenciesof the pipe, the resonant frequency dominates over other frequencies. In that casewe obtain an almost harmonic perturbation. Some overtones will, however, stillbepresent. Using equation 3.9 it is expected that the contributions of the different har-


−2 −1.5 −1 −0.5 0 0.50

100

200

300

400

500

distance from upstream end of turbinemeter (m)

pres

sure

ampl

itude

(P

a)

positionrotor

(a) 24 Hz

−2 −1.5 −1 −0.5 0 0.550

100

150

200

250

300


pres

sure

am

plitu

de (

Pa)

positionrotor

(b) 730 Hz

Figure 3.7: The pressure amplitude of the standing wave in the set up. The dots representthe measured pressure amplitude of the pressure transducers. Figure (a) showsa example of a measurement at 24 Hz, (b) shows a measurement at730 Hz.Both measurements are carried out at a mainstream velocityu0 = 2 m/s. Thedifferent lines indicate the three different parts of the acoustic model (figure 3.4).

monics add quadratically to the error:

Epuls =1

2

[

( |u′1|uin

)2

+

( |u′2|uin

)2

+

( |u′3|uin

)2

+ .....

]

(3.21)

where the subscripts indicate the different harmonics.This is checked by inducing pulsations using the loudspeaker and the sirensi-

multaneously at different frequencies. As is shown in table 3.2 the measurementerror caused by the frequencies separately accumulates,Eadd, within measurementaccuracy to the measurement error caused by the two frequencies simultaneously,Esim. When the difference in frequency is small the induced acoustical velocitywilldisplay low frequency beats which the siren can follow. This produces thetype ofsignal shown in figure 3.8. As the frequency obtained by using the siren displayssome drift in time, we observe some time dependence in the frequency of the beatsfor simultaneous measurements with two frequencies close together. As shown infigure 3.8 att = 20 seconds the loudspeaker is turned on and the turbine meter startsto measure a higher velocity. After another 20 seconds the siren is turned on, themeasurement error becomes larger and starts oscillating. This is by the beats. Takingthe time average of the error during the beats we still find thatEadd ⋍ Esim (table3.2).

For signals in which other frequencies are present, the contribution fromeachfrequency can be added to predict the total error. As the error depends quadraticallyon the amplitude the harmonics with higher amplitude will dominate.


|u′|/u0 |u′|/u0 Eadd EsimEsim−Eadd

Esim× 100%

f = 24Hz f = 164Hz

0.21 0.19 0.032 0.033 0.12 %0.21 0.33 0.074 0.074 -0.18 %0.21 0.57 0.227 0.228 0.64 %

f = 164Hz f = 367Hz

0.09 0.07 0.005 0.005 -3.06 %0.09 0.12 0.011 0.010 -2.73 %0.09 0.16 0.015 0.016 1.04 %

f = 164Hz f = 166Hz

0.26 0.18 0.036 0.040 6.17 %0.26 0.25 0.049 0.048 -1.07 %

f = 164Hz f ≈ 164Hz

0.18 0.28 0.042 0.041 -2.32 %0.18 0.33 0.052 0.056 7.03 %

Table 3.2: Measurements with pulsations at two frequencies. Eadd is the added measurementerror of these frequencies separately andEsim is the measurement error for mea-surement for both pulsations simultaneously. All measurements are carried out ata mainstream velocity ofu0 = 2 m/s.


0 5 0 1 0 0 1 5 0 2 0 0 2 5 01 . 9

1 . 9 5

2

2 . 0 5

2 . 1

2 . 1 5

t i m e ( s )

veloc

ity(m

/s)

l o u d s p e a k e r o n

l o u d s p e a k e r o f f

s i r e n o n

s i r e n o f f

Figure 3.8: Mainstream velocity of the flow measured with theturbine meter. After 20 sec-onds the loudspeaker is turned on at a frequency of 164Hz, after another 20seconds the siren is turned on at a frequency of about 164.2 Hz.

3.4.4 Measurements of velocity pulsation in the field

Determining the amplitude of the velocity pulsations using eight pressure transducersis not practical for industrial use of turbine flow meters. For this reason the optionsto measure the velocity amplitude by means of a hot wire or two pressure transducersembedded in the turbine meter has been investigated.

A hot wire determines the local velocity as well as velocity fluctuations, in theset up two hot wires are placed. Both hot wires are placed 1cm upstream of therotor and 1cm from the pipe wall in the flow. One hot wire was placed in the wakeof a vane of the flow straighteners and the other was placed in between two vanes(see figure 3.3). This was done to account for the effects of the complicated flowprofile around the flow straightener. The mainstream velocity measured locally withthe hot wires are higher, than the mean velocity measured by the turbine meter.Thisis caused by hot wire measurements being local measurement, and the flow profile inthe annulus is not uniform. The local flow velocity can be higher or lower than themean velocity depending on the position of the hot wire. As expected the averagevelocity in the wake of the vane is lower than the velocity measured between thevanes of the flow straightener. The measured amplitude of the velocity pulsations forhigh velocities is within 10% of the amplitude of the velocity pulsations determined

3.5. Determination of the measurement error of the turbine meter 65

with the acoustical model, for low velocities, however, they are much less accurate.The velocity amplitude measured with the hot wire placed in the wake measuressystematically a higher velocity amplitude than the other hot wire. This can probablybe explained by the contribution of acoustically induced vortices shedding at the vane.

For standing waves when the upstream wave is equal to the downstream wave,p+ = p−, we have a negligible phase difference, we can apply a linear approximationusing only pressure transducers at two positions in the turbine meter close totherotor. By using the excitation frequency and considering only plane waves, we candetermine the amplitude of the velocity pulsation. Using the linearised momentumequation for harmonic perturbations it follows that:

uac =∆p′

2ρ0πfLm, (3.22)

with ∆p′ the pressure difference andLm the distance between the two microphones.Taking compressibility into account this becomes:

uac =

∣

∣

∣

∣

∆p′

2ρ0πfLme−i π

2 (1 −M2) +M

(

∆p′

ρ0c0

)∣

∣

∣

∣

, (3.23)

where M= u0/c0 is the Mach number. The applicability of this method is very de-pendent on frequency and standing wave pattern, comparable to the situation shownin figure 3.6. Our measurements show that the velocity amplitude is predicted within40%. However, in general we will not observe standing waves. Therefore, the veloc-ity amplitude can be calculated using the acoustical model described in section 3.4.1using just the two pressure transducers in the turbine meter. However, these resultsshow similar accuracy as the standing wave approximation described above.

In general the acoustical signal should be distinguished from the pressure fluctu-ations induced by vortices. For plane waves this can be done by using morethan twomicrophones. Using a few microphones at one specific distance from the rotor onecan find the plane wave contribution by selecting the coherent part of the signal ofthe microphones in that plane.

3.5 Determination of the measurement error of the turbine meter

During a measurement the rotation speed of the rotor is recorded, without flow per-turbations and with flow perturbations generated by the loudspeaker or thesiren. Theeffect on the rotation speed, averaged over one revolution, is determined from a vi-sual examination of the plots of the signals as shown in figure 3.9. Using a rulerdeviations in the signal of 0.1% can be determined.

Epuls =ω − ω0

ω0, (3.24)


0 2 0 4 0 6 0 8 01 . 8

2

2 . 2

2 . 4

2 . 6

2 . 8

3

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0

s i r e n o f f

(a)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 . 8 21 . 8 31 . 8 41 . 8 51 . 8 61 . 8 71 . 8 81 . 8 91 . 9

1 . 9 1

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0



(b)

0 2 0 4 0 6 0 8 01 . 9

1 . 9 0 5

1 . 9 1

1 . 9 1 5

1 . 9 2

1 . 9 2 5

1 . 9 3

1 . 9 3 5

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0

s i r e n o f f

(c)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 . 7 1

1 . 7 2

1 . 7 3

1 . 7 4

1 . 7 5

1 . 7 6

1 . 7 7

1 . 7 8

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0



(d)

0 2 0 4 0 6 0 8 01 . 9 1

1 . 9 1 5

1 . 9 2

1 . 9 2 5

1 . 9 3

1 . 9 3 5

1 . 9 4

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0

s i r e n o f f

(e)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 . 8 1 51 . 8 2

1 . 8 2 51 . 8 3

1 . 8 3 51 . 8 4

1 . 8 4 51 . 8 5

1 . 8 5 51 . 8 6

1 . 8 6 5

t i m e ( s )

veloc

ity (m

/s)

w - w 0

w 0

l o u d s p e a k e r o n l o u d s p e a k e r o f f

(f)

Figure 3.9: The velocity measured with the turbine meter. The black oscillating line corre-sponds to the instantaneous reading of the flow meter. The smooth white linerepresents the measured velocity averaged over one rotation. The left figuresshow typical measurements for perturbations generated by the siren. The sirenis turned on the first 30 seconds and then turned off. The rightfigures show typ-ical results of measurements for the case in which perturbations are generatedwith the loudspeaker, starting with the speaker turned off,then turned on, subse-quently turned off. The figures show some examples of different pulsation levelsfrom extreme high (a,b) to low (e,f). Due to the long transient in the siren theinfluence of low pulsation levels cannot be detected (e) while they are still veryclearly observable when the loudspeaker is used (f).

3.6. Measurements 67

whereω is the angular velocity of the rotor while measuring pulsating flow andω0

is the angular velocity of the rotor for flow without pulsations. The variationsof thepressure in the reservoir, induces slow mass flow variations during a measurement.This is the main cause of inaccuracies in determining the measurement error oftheflow meter.

The siren needs some time to reach a constant pulsation frequency, therefore thesemeasurements are started with the siren already turned on. After the siren is turnedoff, this effect can also be seen. The slowing down of the siren causesthe frequencyto decay, possibly inducing pulsations that can momentarily cause a large oscillationsin the measuring error during the transition. The influence of the pulsations ismoreaccurately determined by using the loudspeaker. The loudspeaker is turned on andoff during the measurement without complex transitional behaviour, making iteasierto measure the effect of the pulsations. However, the loudspeaker couldonly be usedat low flow velocities, up to 5m/s in the main pipe. Measurements carried out withthe siren do match the corresponding measurements carried out with the loudspeaker.

3.6 Measurements

To investigate the effect of velocity pulsation on the flow measurements of the tur-bine meter, measurements were carried out at resonance frequencies of the set upbetween 24Hz and 730Hz and amplitudes of velocity pulsations ranging fromsmall,uac/u0 ≈ 0.01, to very high amplitudes,uac/u0 ≈ 2. The turbine flow meterused in the set up (Instromet type SM-RI-X G250) has a flow range from20 to 400m3/h (5.6 10−3 m3/s to 0.11m3/s), this corresponds to a velocity in the pipe ofu0 = 0.7m/s to 13.3m/s. In our measurements velocities were varied fromu0 =0.5m/s up to 15m/s. In figure 3.10 and in figure 3.11 measurements are shownfor a pulsation frequency off = 164 Hz for different mainstream velocities. Bothfigures show exactly the same data set, however, in figure 3.10 the data is shown ona double logarithmic scale and in figure 3.11 the data is shown on a linear scale.

From these figures it is clear that for a large range of relative velocity amplitudeextending over two decades and the range of main stream velocities, the measure-ments are still in fair agreement with the quasi-steady theory presented in 3.2.1. Weobserve less than 40% deviation from the theory. By looking at the data, wecan seethat the deviation from the quasi-steady theory increases for decreasing main flowvelocities. Data obtained for pulsation frequencies of 24, 69, 117, 360 and 730Hzare shown in Appendix C. In the section below the effect of the Strouhal number andthe Reynolds number will be investigated systematically.


10−2

10−1

100

101

10−5

10−4

10−3

10−2

10−1

100

101

relative acoustic amplitude, |u’|/u0

rela

tive

mea

sure

men

t err

or, E

puls

quasi−steady theoryu0 = 15 m/su0 = 1 m/su0 = 1.3 m/su0 = 1.5 m/su0 = 1.7 m/su0 = 2 m/su0 = 3 m/su0 = 5 m/su0 = 7 m/su0 = 10 m/s

Figure 3.10: The relative measurement error,Epuls, as a function of the relative amplitudeof the pulsations,|u′|/u0, for measurements at a frequency of 164Hz. Plottedusing double logarithmic scale

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


rela

tive

mea

sure

men

t err

or, E

puls

quasi−steady theoryu

0 = 15 m/s

u0 = 1 m/s

u0 = 1.3 m/s

u0 = 1.5 m/s

u0 = 1.7 m/s

u0 = 2 m/s

u0 = 3 m/s

u0 = 5 m/s

u0 = 7 m/s

u0 = 10 m/s

Figure 3.11: The relative measurement error,Epuls, as a function of the relative amplitudeof the pulsations,|u′|/u0, for measurements at a frequency of 164Hz.


3.6.1 Dependence on Strouhal number

In order to verify the range of validity of the quasi-steady theory, measurements havebeen carried out for a wide range of Strouhal numbers, Sr= fLblade

u0, wheref is the

frequency of the pulsations,Lblade is the length of a rotor blade at the tip andu0 isthe main flow velocity at the position of the rotor. It is expected that for low Strouhalnumbers the quasi-steady theory is valid. From figure 3.11 (and AppendixC), itis found that measurements for a given fixed frequency,f , and a fixed mainstreamvelocity,u0, have a quadratic dependence on the relative velocity amplitude,uac/u0.To investigate the dependence of the deviation in measured volume flow and actualflow Epuls, a quadratic function:

Epuls = a

(

uac

u0

)2

, (3.25)

was therefore fitted through the measured data at a given frequency,f , and flowvelocity, u0 using least-square fitting. The parametera will be referred to as the”quadratic fit parameter”. This parameter is1

2 for the quasi-steady theory. An exam-ple is shown in figure 3.12 for measurements at main stream velocityu0 = 1 m/sand at frequency of pulsationf = 164 Hz.

We will consider only measurements with a relative amplitudeuac/u0 < 1to obtain the quadratic dependence, because higher amplitudes no longer show thequadratic dependence. Measurements with relative amplitudesuac/u0 > 1 are dis-cussed separately in section 3.6.3.

In figure 3.13 the quadratic fit parameter,a, for the measurement at pulsationfrequencies of 24, 69, 117 and 164Hz and mainstream velocities from 1m/s to 15m/s are plotted against Strouhal number. The error bar gives the 95% confidencelevel for the quadratic fit through the measured data. It is an indication forthe qualityof the quadratic fit. In figure 3.13 the data measured using the siren are solidsymbols.The trends in the pulsation error measured with the siren and loudspeaker do notdiffer from each other. However around SrLblade

= O(1), the siren data seems to havea slightly higher quadratic fit parameter than the loudspeaker data. An explanationfor this could be that most of the measurements using the loudspeaker are for smallerrelative amplitudes compared to the measurements using the siren. This indicatesthat at low amplitudes the pulsation error,E, is probably not exactly quadraticallydependent on the velocity amplitude.

The figure shows a clear Strouhal dependence, where the deviation from the ac-tual flow decreases with increasing Strouhal number. However, the deviation from theactual volume flow stays within 40% of the quasi-steady theory for Strouhalnumber,SrLblade

, up to 2.5. Using regression, an equation is obtained to predict the depen-


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

relative acoustic amplitude, uac

/u0

rela

tive

puls

atio

n er

ror,

Epu

ls

quasi−steady theorymeasured data

fitted equation, a(uac

/u0)2

Figure 3.12: A quadratic fit,Epuls = a(uac/u0)2 with a = 0.35 is shown for the mea-

surement data at a mainstream velocityu0 = 1 m/s and with pulsations offrequencyf = 164 Hz. Quasi-steady theory givesa = 1/2.

dence ofEpuls on the Strouhal number, SrLblade:

Epuls = −0.3672 Sr1

5

Lblade+ 0.7407 , for 0.05 ≤ SrLblade

≤ 2.5 . (3.26)

This is a purely empirical relationship between the deviation and the Strouhal num-ber, which cannot be explained theoretically. It is interesting to note that for SrLblade

<0.2 we finda > 1

2 . We cannot explain this.In figure 3.14 the data measured at the higher frequencies (f = 360 and 730

Hz) are shown separately, because they display different behaviour compared to thelow frequency data. The measurements at a pulsation frequency of 360Hz all havea quadratic fit parameter,a, around the 0.5. These measurements are closer to thedeviation,Epuls, found by the quasi-steady theory than the equation found for thelower frequencies. The measurements at a pulsation frequency of 730Hz showthe same quadratic fit parameter for Strouhal numbers of around 6. However, at aStrouhal number of about 10 the data seems to support the empirical relationfoundusing the lower pulsation frequencies. It is possible that these frequencies correspondto mechanical resonant frequencies of the turbine meter causing a different behaviourof the rotor.


0 0.5 1 1.5 2 2.50.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

quad

ratic

fit p

aram

eter

, a

SrL

blade

24 Hz69 Hz117 Hz164 Hz

quasi−steady theory

Figure 3.13: The quadratic fit parametera is plotted against Strouhal number for pulsationfrequencies of 24, 69, 117 and 164Hz and mainstream velocities from 1 to15m/s. The error bars represent the 95% confidence level for the quadraticequation fitted through the measured data. The solid line shows the quadraticfit parameter,a = 0.5, for the quasi-steady theory, the dashed line is a functionfitted through the data found from the present measurements.The solid symbolsrepresent the measurements using the siren, the open symbols the measurementsusing the loudspeaker.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SrL

blade

quad

ratic

fit p

aram

eter

, a

360 Hz730 Hz


Figure 3.14: The quadratic fit parametera as a function of Strouhal number for pulsationfrequencies of 360 and 730Hz. The ’♦’ show the data of measurements atpulsation frequencies of 360Hz, the ’�’ show the data of measurements atpulsation frequencies of730Hz. The solid symbols represent the measurementsusing the siren, the open symbols the measurements using theloudspeaker.


Besides this, as explained in section 3.4.3 for high frequencies small errors in thepressure can cause large errors in estimated acoustical velocity. This could explainsome of the differences in Strouhal number dependence. Another possibility is thatthe measurements at these frequencies are difficult is the presence of acoustic reso-nance. When the length of the constriction caused by the core of the meter matchesabout half the wave length of the acoustical waves, there will be a resonance in thispipe segment. The length of this constriction is about 25cm, i.e. this would giveresonance frequencies of 680Hz. For this resonance the rotor is close to a pres-sure antinode which corresponds to conditions in which the acoustical velocity at theturbine is difficult to determine.

To exclude the possibility that the hollow space inside the flow straightener couldact as a Helmholtz resonator, this area was filled with foam. This did not change theobserved response of the flow meter to pulsations.

Care was taken to prevent these problems occurring at high frequencies by ex-cluding measurements for which the rotor was close to a pressure antinode.1 To beable to draw more conclusions for the deviation,Epuls, at Strouhal numbers greaterthan 2.5, additional measurements are necessary.

At high frequencies, 367Hz and 730Hz, other strange phenomena can be found;at low pulsation levels,|u′|/u0 ≤ 0.1. A negative measurement error can be observed(see figure 3.15) for low velocities, up to 2m/s. These errors do not always repro-duce. We suspect here a combination of mechanical vibration and friction.Duringour tests dust particles were present in the flow and this affected the friction in the ro-tor. However, tests after cleaning the rotor indicated that this had only a minoreffecton most of our data. No significant effect is found foru0 > 2 m/s.

3.6.2 Dependence on Reynolds number

To investigate if there is also a dependence of the deviation,Epuls, on the Reynoldsnumber, ReLblade

, the residual of the Strouhal number dependence predicted by theempirical relation (equation 3.26) and the quadratic fit parameter found forthe mea-surements is plotted as a function of Reynolds number, ReLblade

for pulsation fre-quencies of 24, 69, 117 and 164Hz (see figure 3.16).

Figure 3.16 shows no significant correlation, between the Reynolds number andthe difference between the measurements and the empirical relation for the Strouhalnumber dependence. We conclude that that there is no significant dependence of theReynolds number, ReLblade

, on the deviation,Epuls.

1 Note that all measurements between 360Hz and 730Hz have been rejected because of a verylarge scatter in the quadratic fit coefficienta, which was related to difficulties in the measurement of theacoustical velocity.


0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 9 4 60 . 9 4 80 . 9 5

0 . 9 5 20 . 9 5 40 . 9 5 60 . 9 5 80 . 9 6

0 . 9 6 20 . 9 6 40 . 9 6 6

t i m e ( s )

veloc

ity(m

/s)



Figure 3.15: Some measurements at low velocities and high frequency show a negative mea-surement error. In this plot the velocity of the flow measuredby the turbinemeter is given, the smooth white line represents the velocity averaged over onerevolution of the turbine meter. The loudspeaker is turned on after 30s inducingpulsations of 730Hz, after another 30s the loudspeaker is turned off.

3.6.3 High relative acoustic amplitudes

Several measurements were carried out at relative pulsation amplitudes larger thanunity; uac/u0 > 1. Such high pulsation levels are not likely to occur in practice.However, to investigate the range of the applicability of the quasi-steady theory, itis interesting to look at these results. At these high amplitudes the deviation,Epuls,can no longer be described by the quadratic dependence found for lower amplitudes.In general the measured deviation,Epuls, is smaller than the deviation found by ex-trapolation of the quadratic dependence found foruac/u0 < 1. The difference withthis quadratic dependence is still small for relative acoustic amplitudesuac/u0 ≈ 1and increases for increasing amplitude. Typical measurement data is shown in fig-ure 3.17. At pulsation levels,uac/u0 ⋍ 2.5, the quasi-steady theory overestimatesthe effect of pulsations by about a factor 2. While for low amplitudes the effect ofpulsations is overestimated by a factor 1.4.


0 1 2 3 4 5 6 7

x 104

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

ReL

blade

a −

afit

Figure 3.16: The difference between the quadratic fit parameter found for the measured data,a and the quadratic fit parameter found by the empirical relation in section3.6.1,afit,is plotted as a function of Reynolds number for pulsation frequenciesof 24, 69, 117 and 164Hz.

3.6.4 Influence of the shape of the rotor blades

The influence of the shape of the blade was investigated by replacing the standardrotor with a rotor a with different blade shape. The original rotor has blades with arounded upstream leading edge and a chamfered trailing edge. The rotorwas replacedby a rotor with chamfered leading edges similar to the trailing edges (figure 3.18).

To determine the behaviour of the rotor with chamfered leading edges in pulsatingflow some of the measurements carried out with the standard rotor are repeated usingthe new rotor. Figure 3.19 shows the results of the measurements carried out ata pulsation frequency of 164Hz and mainstream velocities ofu0 = 1, 5 and 15m/s, compared to the measurement data obtained for the standard rotor. Withinthe accuracy of the measurement no difference was found. To verify this further aquadratic fit as explained in section 3.6.1 was made and this parameter was plottedagainst Strouhal number, SrLblade

for low frequencies (f = 24, 69, 117 and 164Hz)(figure 3.20). Again, we see that within the accuracy level of the measurements thereis no difference between the deviation of the volume flow measurement for therotorwith blades with rounded leading edges and the rotor with blades with chamferedleading edges.

3.7. Conclusions 75

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

relative acoustic amplitude, uac

/u0

rela

tive

puls

atio

n er

ror,

Epu

ls

quasi−steady theorymeasurement data

fitted equation, a(uac

/u0)2

Figure 3.17: The deviation,Epuls as a function of the relative acoustic amplitude,uac/u0. Aquadratic fit,Epuls = a(uac/u0)

2 derived foruac/u0 < 1 (with a = 0.35) isshown for the measurement data at a mainstream velocityu0 = 1m/s and withpulsations of frequencyf = 164 Hz.

3.7 Conclusions

The effect of the pulsating flow on a turbine flow meter has been investigatedexper-imentally and results have been compared to the results of a simplified quasi-steadymodel. A set up was built making it is possible to induce pulsations with a frequencyfrom 24Hz to 730Hz, relative acoustic velocity amplitudes,uac/u0, from2×10−2

up to 2 and volume flows ranging from the minimum to the maximum flow specifiedby the manufacturer, i.e. from 20 to 400m3/h. Multi-microphone measurementshave been used to determine the amplitude of the velocity pulsation at the rotor. Theerror caused by the pulsations is obtained from the comparison of the rotation speedof the rotor in presence of pulsations with the one in the case that there are no pul-sations. The measurements show that the simplified quasi-steady theory gives a fairapproximation of the error caused by the pulsations. The measurements agree withthe theory within 40% for nearly all measurements, even for measurements athighrelative acoustical amplitudes. We found that the error caused by pulsations is de-pendent on Strouhal number. For SrLblade

< 2.5 an empirical relation was found forthe dependence of the error on the Strouhal number. Globally one expects that theinfluence of pulsations should decrease with increasing pulsation Strouhal number.


(a) rounded leading edge (b) chamfered leading edge

Figure 3.18:A schematic drawing of the rotor (a) with rounded leading edges and (b) with achamfered leading edges used in the measurements

This corresponds to our observations. As of yet no physical explanation is found forthis specific dependence. For SrLblade

> 2.5 the behaviour of the rotor is still unclear,caused by the difficulties in measuring at higher pulsations frequencies. The measure-ment error caused by the pulsations is not significantly dependent on the Reynoldsnumber. The shape of the upstream edge of the rotor blades does not influence theStrouhal number dependence of the systematic error induced by the pulsations.

This study stresses the importance of determining the acoustical velocity at therotor for a correction of measurement errors due to pulsations. Measurements withlocal velocity probes such as hot wires are difficult to use because theydo not distin-guish between vortical perturbations and acoustical waves. Acousticalwaves can bedetected by means of microphones mounted flush in the wall. This would howeverinvolve multiple microphones at a certain position to allow the detection of the planewaves by cross-correlation method analogous to microphone array techniques.

3.7. Conclusions 77

0 0.1 0.2 0.3 0.4 0.5 0.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


rela

tive

mea

sure

men

t err

or, E

puls


0 = 1 m/s; round l.e.

u0 = 1 m/s; sharp l.e.

u0 = 5 m/s; round l.e.


u0 = 15 m/s; round l.e.


Figure 3.19: The relative measurement error,Epuls, is plotted against the relative pulsa-tion amplitude,|u′|/u0, for measurements with the ’new’ rotor with chamfered-leading-edge blades and the standard rotor with rounded-leading-edge blades.The data is for measurements at a pulsation frequency of 164Hz and main-stream velocities,u0 = 1, 5 and 15m/s. The solid symbols are the data mea-sured with the rotor with chamfered-leading-edge blades. The plot shows thatwithin the measurement accuracy there is no difference in behaviour for the tworotors.

0 0.5 1 1.5 2 2.50.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

quad

ratic

fit p

aram

eter

, a

SrL

blade


Figure 3.20: The quadratic fit parametera is plotted against Strouhal number for pulsationfrequencies of 24, 69, 117 and 164Hz and mainstream velocities from 1 to15m/s. The ’o’ represents the measured data of the standard rotor with bladeswith rounded leading edges and ’*’ represents the data measured with the ’new’rotor with blades with chamfered leading edges. Again, no difference can befound within measurement accuracies for the two rotors.


4

Ghost counts caused by pulsations withoutmain flow

4.1 Introduction

Turbine flow meters are often placed in measurement manifolds, consisting ofseveralruns (side branches). At low volume flows some of the branches in these manifoldsare closed. Closed side branches can form resonators which are driven by vortexshedding at the junction with the main pipe (Peters, 1993). Flow pulsations in thesemanifolds can then affect turbine meters in open pipes, but can also induceghostcounts. This corresponds to spurious flow measurement of the turbine meter in closedside branches where there is no main flow. These ghost counts start above a criticalpulsation level.

The aim of this chapter is to obtain a better understanding of these spuriouscounts. In the first part of this chapter1 a theoretical model is presented whichexplains the occurrence of these spurious counts in the limit of infinitesimally thinturbine blades. The predicted threshold for the occurrence of spurious counts is com-pared to experimental data at various gas pressures in the range from 1to 8 bar.In the second part of this chapter, a numerical and experimental study ofthe flowaround the edge of a turbine blade is presented. Aim of this study is to predict theinfluence of the thickness and the shape of the turbine blade on the onset of spuriouscounts. An experimental setup has been built to simulate the flow around a model ofa blade edge. The vortex shedding at the edge has been visualised. On the blade thesurface pressure has been measured. These measurements have been compared withpredictions of a discrete vortex model.

1 The first part of this chapter has been published with some minor changes in the Journal of Fluidsand Structures; P.W. Stoltenkamp, S.B. Araujo, H.J. Riezebos, J.P.Mulder and A. Hirschberg (2003),Spurious counts in gas volume flow measurements by means of turbine meters, 18(6):771-781.

80 4. Ghost counts caused by pulsations without main flow

4.2 Onset of ghost counts

4.2.1 Theoretical modelling of ghost counts

The spurious count behaviour of a turbine meter can be explained by considering theforces acting on an aerofoil in an oscillating flow. The blades of the turbinerotormost commonly used in gas transport systems have a rounded leading edge and asharp trailing edge (see figure 4.1). This difference in edge shape causes the spuriousrotation.

L b l a d et b l a d e

u '

l e a d i n g e d g e

t r a i l i n g e d g e

Figure 4.1: Blades of the rotor

In our model it is assumed that there is only flow separation at the sharp trailingedge of the blade. The flow separation at the sharp edge in an oscillating flow canbe seen in Schlieren visualisations (see figure 4.2). In the following we neglect theinteraction between the blades in the rotor.

Centrifugal forces in a potential flow around the edge of a plate cause a lowpressure at the surface of the blade edge, which results in a force directed along theblade, a ”suction force”, which will be called the edge force. In case ofa sharp edgevortex shedding reduces this edge force, while for a rounded edge it remains presentas long as the flow remains attached. This leads to a net force on the blade. Thisforce brings about a torque on the rotor. Spurious counts start when this torque islarge enough to compensate the torque due to the static friction forces. This analysisis restricted to the case of a harmonic acoustical oscillation with frequencyf (Hz)

4.2. Onset of ghost counts 81

Figure 4.2: Schlieren visualisations of the flow separationat the sharp edge at a Strouhalnumber,Srtblade

= O(1)

and amplitudeuac (ms−1) of the particle velocity:u′ = uac cos(2πft). Furthermore,the turbine is considered at the condition that it does not yet rotate, so thatthe bladeshave a fixed position.

Important dimensionless numbers for this problem are the Helmholtz number,He = ftblade/c , the Strouhal number,Srtblade

= ftblade/uac, the Reynolds number,Re = uacLblade/ν and the ratio,tblade/Lblade, of the thickness,tblade, of the bladecompared with the length,Lblade. In these numbersν (m2s−1) is the kinematicviscosity andc (ms−1) the propagation speed of sound waves.

It is assumed that the flow is attached and that the viscous boundary layersare thin(Re >> 1). Hence, the flow around a blade of the turbine meter can be describedwith potential flow theory corrected for the effect of boundary layers.The flow isassumed to be locally incompressible, because the rotor is small compared to theacoustical wave length and the amplitude,uac, is small compared to the speed ofsound (He << 1 andM = uac/c << 1). The Strouhal number,Srtblade

gives anindication of the blade thickness compared to the acoustical displacement of particles.If the Strouhal numberSrLblade

= SrtbladeLblade/tblade is small, the blade length is

small compared to the acoustical particle displacement and the flow can be assumedto be quasi-steady. If the Strouhal number is of order unity,Srtblade

= O(1) withtblade/Lblade << 1, local vortex shedding occurs at the edges of the blade. Finally,if the Strouhal number is much larger than unity,SrLblade

>> 1, vortex sheddingis negligible except for very sharp edges. In that case the blade thickness is not therelevant length scale. This is illustrated in figure 4.3.

In practice the ratio,tblade/Lblade, of the thickness of the blade compared to the


Sr tblade = O (1), t blade / L blade = O (10 - 1 )

Sr tblade >> 1 , t blade / L blade = O (10 - 1 )

Sr tblade L blade /t blade <<1

u'

Figure 4.3: Influence of Strouhal number and blade length to thickness ratioL/tblade on theflow. SrLblade

≪ 1 corresponds to quasi-steady flow. WhenSrLblade> 1, but

Srtblade< 1 we have strong vortex shedding at the edges. ForSrLblade

≫ 1 andSrtblade

> 1 we have local vortex shedding at the sharp edges.

length, is small,tblade/Lblade = O(10−1). In this theory, the blade is modelledas a flat plate. In a first model the flow separation at the sharp edge is modelledby means of a single point vortex and by applying the Kutta condition (also calledKutta-Joukowski condition) at the sharp edge. This corresponds to themodel ofBrown and Michael (1954) for flow separation at the leading edge of a slender deltawing. The force on the blade is found by integration of the pressure distribution onthe plate. The singular flow around the sharp leading edge results in a finite so-callededge force which is the result of an infinite low pressure applied on a zerosurface(Milne-Thomson, 1952). A second model is considered for the limit case, for whichthe Strouhal number is very large,Srtblade

>> 1. Here a potential flow is consideredwithout flow separation, but the contribution of the sharp trailing edge to the nethydrodynamic force on the blade is removed. The idea is that the vortex sheddingat this sharp edge has removed the local flow singularity without affecting the globalflow around the blade.

Theory for the caseSrLblade= Srtblade

(Lblade/tblade) = O(1)

The theory for the case in which the Strouhal number is order unity is consideredfirst. In a point vortex model, the vortex sheet generated by flow separation at thesharp edge is represented by a single point vortex of varying circulation. The vortexis assumed to be connected to the sharp edge of the plate by means of a feeding sheet.The circulation of the point vortex is calculated by applying the Kutta condition at thesharp edge. The Kutta condition requires the velocity to be finite at the sharpedge.In a real flow, this means that the flow leaves the edge tangentially, accounting forviscous effects. In a point vortex model this implies that at the edge a stagnation pointis assumed. The point vortex moves with the flow. Application of the Kutta condition


implies then that the circulation changes with time. The convection velocity of thepoint vortex is calculated by means of potential flow theory. For this calculation afree vortex is assumed. This assumption is in contradiction with the time dependenceof the circulation of the vortex. This induces a spurious force that will be neglectedfurther (Rott, 1956). This error will appear not to be critical for our results.

α Im[ z ]

Γ v

Re[ z ]

Γ v

α

Re[ ξ ]

Im[ ξ ]

A

z -plane ξ -plane

u'

Figure 4.4: Flat plate in the z-plane and transformed to a circle in theξ-plane

The flow potential is calculated using a complex potential and conformal map-ping. A circle with radiusA, in theξ-plane, is transformed in a flat plate of lengthLblade = 4A, in thez-plane using the transformation of Joukowski:

z = ξ +A2

ξ, (4.1)

The complex potential of the flow,Φ, in the transformed plane can now be writtenas:

Φ = u′ξe−iα +u′A2

ξeiα − iΓv

2πln(ξ − ξv) +

iΓv

2πln(ξ − A2ξv

|ξv|2) . (4.2)

whereα is the incidence angle of the flow with respect to the blade (see figure 4.4).2

The first and the second term on the right hand side are the acoustic flow poten-tial for a circular cylinder in a parallel flow after applying the Milne-Thomsoncircletheorem (Milne-Thomson, 1952). The third and the fourth term are the contributionsof the vortex and that of the mirror-imaged vortex atξ = A2/ξ∗v , also found using

2 Note that using a two-dimensional infinite cascade representation of the rotor we could use con-formal mapping from a cascade to a circle proposed by Durant (1963). This would allow to take theinteraction between the rotor blades into account.


Milne-Thomson’s circle theorem. Hereξv is the position of the vortex in the trans-formed plane andΓv the circulation of the vortex. The circulation of the vortex iscalculated using the Kutta-condition atξ = A:

dΦ

dξ

∣

∣

∣

ξ=A= 0 . (4.3)

At the first step the vortex is shed, the position of this vortex is calculated us-ing the self-similar solution, given by Howe (1975) for an impulsively startingflowaround a semi-infinite plate. The velocity of the vortex,uv, is calculated in the fol-lowing steps using the following equation, assuming the vortex is a free vortex:

u∗v =dz∗vdt

=dΦ

dz

∣

∣

∣

z=zv

limξ→ξv

[

dΦ

dξ+

iΓv

2π(ξ − ξv)

]

/dz

dξ+

iΓv

2π

d2z

dξ2

/

(

dz

dξ

)2

,

wherezv is the position of the vortex,ξv the position of the vortex in the transformedplane,∗ indicates the complex conjugate,Φ the complex potential andΓv the circula-tion of the vortex. The last term is known as the Routh correction (Clements, 1973).The new position of the vortex is calculated using a fourth-order Runge-Kutta inte-gration scheme (Hirsch, 1988). The new circulation,Γv, of the vortex correspondingto this new position, is calculated using the Kutta condition (equation 4.3). In thismodel the circulation of the vortex vanishes when the acoustic flow is zero. At thenext time step a new vortex is shed. As an example the path of a single point vortexand its circulation is plotted in figure 4.5 for a typical case under conditions for whichspurious counts were measured.

0.45 0.5 0.55−0.05

0

0.05

x/L

y/L

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

t/T

Γ/(U

acL

)

Figure 4.5: a) Calculated path of a single point vortex and b)calculated vortex strength forSrLblade

= 9


To calculate the torque on the turbine, the force on each blade is calculated nu-merically by integration of the pressure along the plate.

F = −∮

p · ndS = −∫

without edgepdS −

∫

edgepdS , (4.4)

wherep (Pa) is the pressure andS (m2) the surface area. The pressure,p, in thefirst term is obtained from the potential flow solution using the Bernoulli equation forunsteady potential flow:

ρ∂Φ

∂t+

1

2ρu2 + p = constant(t) . (4.5)

The integration is carried out by means of the midpoint quadrature rule. Forthesecond term, around the singularity, a quasi steady-approximation is used(Milne-Thomson (1952) and Appendix D). As input for this theory a Taylor expansion ofΦaround the pointξ = −A is used.

Limit of the theory for SrLblade= Srtblade

(Lblade/tblade) >> 1

The limit of this theoretical model is considered next forSrLblade>> 1. In such a

case the effect of the vortex on the global flow around the blade is negligible exceptfor the flow near the sharp trailing edge. A great advantage of this simplifiedmodel isthat an explicit expression is obtained for the aerodynamic torque on the rotor withoutthe need to determine the details of the vortex path. As explained above the keyofthis model is that it is assumed that there is only flow separation at the sharp trailingedge of the flat plate. As a consequence, there is a finite velocity at this sharp edgeand therefore no edge force. At the leading, rounded edge there is noflow separationand the velocity becomes infinite and this results in an edge force (figure 4.6).

F e F ea a

Figure 4.6: Flow separation at the sharp edge for a flat plate in an oscillating flow


Because the velocity becomes infinitely large at the edge, convective flow ac-celeration is larger than the local time dependent flow acceleration. A quasi-steadyapproximation can be used. The edge force is directed parallel to the plate and can becalculated with potential flow theory (Milne-Thomson, 1952).3 The magnitude,Fe,of the edge force is:

Fe = −πρSbladeu2ac sin2 α , (4.6)

whereFe (N is the edge force),Sblade (m2) is the surface area of the plate,uac

(ms−1) is the acoustic velocity amplitude andα is the angle between the blades andthe direction of the acoustical flow. The flow separation generates a vortex close tothe sharp edge. As explained above this vortex is assumed to have solely theeffectof removing the edge singularity in the flow field and as a consequence the edgeforce. Flow separation also implies that the boundary layer vorticity is injectedintothe main flow. This vorticity is assumed to be of small magnitude and confined to aregion close to the edge, therefore there will be no significant change ofthe globalcirculation for the flow around the flat plate. If the flat plate is placed in a parallel har-monically oscillating flow, the flow will alternate between the left and right situationin figure 4.6. The force perpendicular to the flat plate will also alternate harmonically.If a harmonically oscillating flow is imposed, the average of the normal force takenover one oscillating period will be zero. Consequently, the resultant averaged forcefor the flat plate over one acoustic period can be simply calculated using the edgeforce. With this edge force, it is possible to calculate the average torque,Tav, on theblades.

Tav =πρravSblade

8u2

ac sin3 α , (4.7)

whererav (m) is the average radius at which the force is applied on the blade.

Comparison the results of the models

In figure 4.7 the relative difference,(T1 − T2)/T2 between the critical torque,T1,calculated with the point vortex model forSrLblade

= O(1) and the critical torque,T2, calculated with the model for the limit caseSrLblade

>> 1 is plotted againstthe reciprocal Strouhal number,1/SrLblade

= uac/fLblade, based on the length ofthe plate. For typical Strouhal numbers as encountered in our experiments(4 <SrLblade

≤ 20) the difference between the results of the two theories is less than 35%which is negligible compared to the difference between theory and experiments.

3 Again in this case the influence of the interaction between the rotor blades canbe taken into accountby considering an infinite two-dimensional cascades of thin plates (Durant, 1963).


- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

0 . 2

0 . 2 5

0 . 3

0 . 3 5

0 . 4

0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 31 / S r L

(T2-T

1)/T

2

Figure 4.7: The difference(T1 − T2)/T2 between the critical torque calculated with the twomodels plotted against the reciprocal Strouhal number

4.2.2 Experimental setup for ghost counts

Acoustical oscillations in a closed side branch can either be induced by a resonantresponse to compressor pulsations or by flow induced pulsations due to vortex shed-ding. In the experimental set up these flow oscillations are induced using a loud-speaker mounted within a closed pipe segment (figure 4.8).

Two set ups were used at Gasunie (Mulder, 2000), a small set up at atmosphericpressure with a pipe diameterD = 100mm, and a large set up, with a pipe diameterD = 300mm. In the large set up we had the possibility to vary the mean static pres-sure from 1 to 8bar. In the small set up, the gas turbine meter (Instromet SM-RI-XG250, see table 2.1) is placed between two PVC pipes with diameterD = 100 mmand lengthLp1 = Lp2 = 1.8 m. A loudspeaker (Visaton W100S) is placed at theend of the pipe, while the other end is closed by a rigid plate. Four dynamical piezo-electric pressure transducers (Kistler type 7031) are placed at positions distributedalong the pipes. The pressure transducers are connected to charge amplifiers (Bruel& Kjaer type 2635). Experiments with the small turbine meter were repeated at Eind-hoven University of Technology (TU/e) with the set up described in section 3.3. Inthe large set up the gas turbine meter (Instromet SM-RI-X G2500) is connected withtwo pipes with diameterD = 300 mm and lengths ofLp1 = 6 m andLp2 = 2 m.The end of each pipe is sealed to be able to support a pressure up to8 bar above at-mospheric pressure. In the long pipe,Lp1, a loudspeaker (Peerless XLS10) is placed.The short pipe,Lp2, is closed by means of a flat and rigid plate. The position of theloudspeaker can be changed, making it possible to modify the resonance frequency of


loudspeaker

field conditions

laboratory experiment main flow

closed side branch

D

L p2 L p1 x=0 x

Figure 4.8: Field conditions compared with experimental set up

the system. Nine holes are made for placing dynamical pressure transducers (Kistlertype 7031) along the pipes, allowing an optimisation of the choice of the position ofthe four available transducers.

To obtain a harmonic voltage signal, a signal generator (LMS Roadrunner) isconnected to a power amplifier (Bruel & Kjaer type 2706) from which the signal isapplied to the loudspeaker. The amplitude and the frequency can be adjusted sepa-rately. For the acquisition of all the signals a twelve channel data acquisition device(LMS Roadrunner compact) is used. The rotational frequency of the blades is mea-sured using standard Elster-Instromet measuring equipment.

The pressure transducers are used to measure the acoustic pressureamplitude.As the acoustic waves in the pipe are plane waves, the pulsation pressure amplitude,p′(x, t), depends only on the coordinate along the axis parallel to the pipe. This canbe assumed if the frequency,f , of the pulsations is much smaller than the cut-offfrequency,fc, for non planar waves (in the small set upfc = c0/(2D) ≈ 1.7 × 103

Hz and for the large set upfc = c0/(2D) ≈ 5.7× 102 Hz). The pressure amplitudedata is used to calculate the acoustic velocity amplitude,uac, if a full reflection at theclosed end wall is assumed and thermal and friction losses in the pipe are neglected.The acoustic velocity at a distancex from the end of the wall of the pipe is calculated


using the equation:

|u′(x)| =|p′|ρ0c0

sin kx

cos(kxm), (4.8)

where|u′| = uac (ms−1) is the acoustic velocity amplitude,|p′| (Pa) is the acousticpressure amplitude measured at a distancexm (m) from the end of the pipe wall,ρ0

(kgm−3) is the density of the propagation medium at ambient temperature,c0 ≈ 344ms−1 is the acoustical wave propagation velocity in air at ambient temperature,k(m−1) is the wave number andxm (m) is the distance between the end wall of thepipe and the pressure transducer. To obtain the acoustic velocity,uac, at the blades ofthe turbine flow meter, incompressibility is assumed and from the mass conservationequation the following is obtained:

uac = |u(Lp2)|Sm

St, (4.9)

whereuac (ms−1) is the amplitude of the acoustic velocity at the turbine blades,Lp2

(m) is the pipe length (see figure 4.8),Sm = πD2/4 (m2) is the surface area of thepipe where the measurement is taken andSt (m2) is the cross sectional surface areaof the turbine flow meter. The flow through the turbine meter can be assumed incom-pressible, because the length of the turbine meter,Lt, is very small compared to thewave length,λ (Lt ≪ λ). The frontal area of the blades of the turbine meter, whichis about 10% of the internal area of the turbine meter, is neglected in calculating St.We used the valuesSt = 5.7 × 10−3 m2 for the small meter (Instrometer G250) andSt = 4.9 × 10−2 m2 for the large meter (Instromet G2500). Measurements ofuac

obtained from the four different pressure transducers agree with each other within10%.

4.2.3 Experiments

Critical friction torque

Two different experiments have been carried out to obtain the critical statictorqueabove which rotation occurs: a dynamic experiment and a static experiment. Theequation of motion for the turbine flow meter is:

Irotordω

dt= Td + Tf , (4.10)

whereIrotor (kg m−2) is the moment of inertia of the rotor relative to its axis,ω(rad s−1) is the angular velocity,t (s) the time,Td = Tav (kg m2s−2) is the drivingtorque at the blades of the rotor andTf (kg m2s−2) the total friction torque. The


approximate driving torque has been derived in the previous section (see equation4.7).

The torque caused by friction can be split into the contribution of air friction,Tair, on the rotor and in the torque caused by the friction,Tmech, in the bearing (thefriction in the oil and the friction of the shaft). The friction of the shaft can be dividedin static friction and dynamic friction.

To obtain the torque due to the dynamic friction, the rotor is accelerated to asteady velocity by means of an acoustic field generated by a loudspeaker.Whenthe loudspeaker is turned off, the decay of the angular frequency is registered and afourth-order curve is fitted through the data. Subsequently, the torque iscalculatedfrom the angular velocity using the relation in equation 4.10. The driving torque,Td

is assumed to have a second order dependency on the angular velocity (see equation4.6). The friction torque of the oil in the bearing has a linear dependency on therotational velocity . The friction of the shaft is assumed to be constant. In thelimit ofvanishing rotational speed, the only contribution to the torque is the constantdynamicfriction of the rotor of the shaft. The value of this dynamic torque is found bytakingthis limit to be approximately6 × 10−6 Nm for the small set up and9 × 10−5 Nmfor the large set up.

The static friction torque is measured using a small piece of adhesive tape fixedat a known radius on the rotor. The tape induces a torque on the rotor, which can becalculated from the weight and the position of the tape. The rotor is restrained andreleased using a photograph shooter. The torque is increased step by step until therelease of the rotor induces rotation. With this method the critical static torque wasfound to be5.6 × 10−6 Nm for the small set up and1.0 × 10−4 Nm for the largeapparatus. These values of the critical static torques agree within the experimen-tal accuracy with the dynamic friction torques in the limit of zero rotational speed.Furthermore, these results agree with the specifications of the manufacturer.

Critical acoustic velocity

Above a critical value of the amplitude of the acoustic velocity,uac, spurious countsoccur. This critical acoustic velocity is measured, by keeping the acoustical excitationfrequency,f , constant, while increasing the amplitude slowly. This can be doneeither manually or by using a special function on the signal generator. When therotor starts to rotate the acoustic velocity is determined from the acoustic pressureamplitude,p′, as explained in the previous section. These results show considerabledeviations for different frequencies, but also for two consecutive measurements at thesame frequencies. The standard deviation is approximately 20% of the mean value.Probable reasons for these deviations can be the varying static friction torque causedby the unevenly distributed oil in the bearing and local roughness of the solid surfaces


in the bearing. Mechanical vibrations induced by the loudspeaker can also influencethe threshold for auto-gyration. It is also possible that they are caused by changesthat occur in the flow topology or by difficulties in the determination of the thresholdof rotation from the experimental data.

The mean critical acoustic velocities are determined for the small and the largeset up at their first and third resonance mode. For the small set up these frequenciesaref1 = 60 Hz andf3 = 210 Hz respectively, and for the large set upf1 = 30 Hzandf3 = 100 Hz respectively. The critical acoustic velocity amplitude,uac, for thelarge set up has been measured at four different static pressures, 1bar, 2 bar, 4 barand 8bar.

4.2.4 Comparing measurements with results of the theory

The results of the experiments can now be compared to the calculated data andto thevalue of the critical pressure amplitudes in field conditions.

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000 6000 7000

Reynolds number (Re Lblade =u ac L blade / )

Str

ouha

l num

ber

(Sr

tblad

e =t b

lade

f/u a

c )

Gasunie, large app. 30 Hz


Gasunie, small app. 60 Hz


TU/e, small app. 69 Hz


ν

Figure 4.9: The critical Strouhal number plotted against the Reynolds number. The dataobtained at Gasunie (grey and solid circles and squares) complemented with thedata obtained at TU/e (open circles and squares) as described in section 4.3).

Figure 4.9 shows the critical Strouhal number,Srtblade= tbladef/uac, based on

the blade thickness,tblade, at which the rotation starts, plotted against the Reynoldsnumber based on the blade length, Re= uactblade/ν, with ν the kinetic viscosity.The critical Strouhal number,Srtblade

, is of order unity, which corresponds to ourqualitative model of the effect of the blade thickness. When the vortex remains at


distances from the edge smaller than the blade thickness, it is not expected that thisaffects the flow. Hence there is no rotation forSrtblade

>> 1. A significant depen-dency ofSrtblade

on the Reynolds number can be observed, which we cannot explainwith the used model. We did actually not expect such a dependency.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1000 2000 3000 4000 5000 6000 7000

Reynolds number (Re Lblade = u ac L blade / )

ratio

of m

easu

red

and

pred

icte

d

criti

cal t

orqu

e







ν

Figure 4.10: Ratio between the measured critical torque andthe calculated critical torqueplotted against the Reynolds number. The data measured at Gasunie (grey andsolid circles and squares) complemented with the data measured at TU/e (opencircles and squares) which are described in section 4.3.

In figure 4.10 the ratio of the measured critical static torque and the predictedtorque calculated using the limit case model (SrLblade

>> 1) for both set ups isplotted against the Reynolds number, with the length of the blade as the characteristiclength. For the small set up the calculated critical torque is two times the measuredcritical torque for the first resonance mode and five times the measured critical torquefor the third mode. For the large set up the calculated critical torque is 20% lowerthan measured for the first mode and 1.4-1.7 times larger for the third mode. Itwouldbe interesting to investigate whether these difference in behaviour for the first andthird acoustical mode of the pipe are related to difference in mechanical vibrationlevel. Such vibrations induced by the loudspeaker could affect the critical torque.

Using the results the critical torque can be estimated for field conditions of naturalgas transportation with a static pressure,p = 60 bar = 6 × 106 Pa and a density ofthe natural gas,ρg = 14 kg m−3. The geometries of the pipe and of the turbine flowmeter are assumed to be similar to that of the large set up. Using equation 4.7 theacoustical velocity,uac, is calculated, at which the critical static torque is reached andspurious counts occur. This is found to beuac ≈ 7 × 10−2 ms−1. This corresponds

4.3. Influence of vibrations and rotor asymmetry 93

to acoustical pressure amplitudes of the order ofp′ = ρ0c0uac ≈ 103 Pa with fornatural gasc0 ≈ 390 ms−1. In field conditions measurements show spurious countswith acoustic pressure amplitudes of4× 103 Pa (Riezebos et al., 2001). This showsthat the model can provide a fair indication for conditions at which spuriouscountsoccur.

4.3 Influence of vibrations and rotor asymmetry

4.3.1 Vibration and friction

To verify the measurements described in section 4.2.3 the same measurements werecarried out in the set up built at Eindhoven University of Technology to measure theeffect on the turbine meter of pulsation with main flow (section 3.3). By closing thevalve and using the loudspeaker to induce pulsation, comparable measurements ofthe critical acoustical velocity amplitude,uac, can be preformed. The turbine meterin this set up is of the same type (Instromet SM-RI-X G250) as used in the smallsetupdescribed in section 4.2.2. In this set up, however, the loudspeaker is mechanicallydisconnected from the pipe so that mechanical vibrations are reduced.

The amplitude of the critical acoustic velocity is determined at different frequen-cies by varying the amplitude step by step and waiting a few moments to observewhether or not the rotor starts rotating. Around the critical amplitude the situationcan occur that the rotor begins to move but stops before a full revolution iscompleted.This shows the dependence of the static friction torque on the rotor position.The dy-namic friction torque was measured using the dynamical method described in section4.2.3 and was found to be1.5 × 10−5 Nm.

The results are plotted in figures 4.9 and 4.10, and show a good agreementwiththe other measurements. It should be noted that is found from this test and acheck bythe manufacturer that the there is an increase of friction caused by the deterioration ofthe bearings. This is caused by some corrosion problem within the pressure reservoir,causing pollution of the flow with rust particles. The increased friction causes a needfor higher pulsation to compensate the friction torque, the maximum pulsation isinduced corresponding with a Strouhal number of Srtblade

= 0.048, using equation4.7 of the limit model, this is equal to a torque of aboutTd = 2.8 × 10−4 Nm.

4.3.2 Rotor blades with chamfered leading edge

In order to verify that the difference in shape of the edges of the bladesof the rotorcauses the ghost counts, the rotor in the turbine meter was replaced with a rotor witha blade profile with a chamfered leading edge (figure 4.11).

Again pulsations were induced to investigate the occurrence of ghost counts. As


(a) rounded leading edge (b) chamfered leading edge

Figure 4.11: A schematic drawing of a rotor (a) with rounded leading edges and (b) withchamfered leading edges used in the measurements

expected, no rotation of the rotor occurred. Also imposing an initial rotation of therotor is not sufficient to induce ghost counts. An initial rotation was induced by amain flow, but when the main flow was stopped the rotor rotation decayed to zeroindependently of the imposed acoustical pulsations.

4.4 Flow around the edge of a blade

In order to get a more quantitative prediction of the threshold for auto-gyration theinfluence of the blade thickness and shape of the edge should be investigated further.The influence of the vortex shedding on the flow around the edge will be examinedmore closely in the present section.

4.4.1 Numerical simulation

To model the vortex shedding from the blade edge the so-called ”vortex-blob” method(Krasny, 1986) has been used. This method, solves the vorticity-transport equationfor two-dimensional flow neglecting viscous effects. For a more detailed descrip-tion of this method we refer to Hofmans (1998) and Peters (1993). The ”vortex-blob” method requires the specifications of separation points. We can use this inorder to identify the contribution of various vortices to the edge force. Thevortic-ity field is modelled by vortex blobs, desingularised point vortices, and by applyingthe Kutta condition at the separation points new vortex blobs are introduced.Thedesingularisation of the vortex is used to avoid numerics-induced chaotic behaviourin vortex-vortex interaction. When considering the interaction of a vortex with a wall

4.4. Flow around the edge of a blade 95

we consider the vortex as a point vortex. The wall is represented by means of a panelmethod. The dipole distribution on the walls is determined such that the normal flowvelocity equals zero at the panel centers.

4.4.2 Experimental set up for flow around an edge

To verify the numerical method described above, an experimental set up was build tostudy the local flow around the edge of a single blade of blade thicknesstblade = 1.00cm. This was done by building a wooden box of 48.2cm x 24.8cm x 22.3cm with awall thickness of 2.5cm divided in the middle to create two separate compartments.These two compartments are connected by means of a duct creating two connectedHelmholtz resonators (see figure 4.12). The volumeV1 of the first compartment isV1 = 8.15×10−3 m3. The volumeV2 of the second compartment isV2 = 8.35×10−3

m3. This implies a small asymmetry in the volumes of the compartments in our setup. The duct cross-sections areS = d x w with d = 4.5 cm andw = 10.0 cm.The edge of the dividing wall is a model for the edge of the turbine blade. The tipof the edge of is placed at a distance of 5.0cm from the top wall closing the setup. It is possible to change the shape of the blade edge. The first model has a sharpchamfered edge with a bevel angle of45◦. The second model has a rounded edgeand the last model has a square edge. The experiments for the rounded edge show novortex shedding at the edge. We present here only the results for the sharp edge. Theinternal dimensions of the set up are given in figure 4.12.

The acoustic flow is determined by using two piezo electrical pressure transducers(PCB 116A), one on each side of the box. The signal of these pressure transducers isamplified be using a charge amplifier (Kistler type 5011). A third miniature pressuretransducer (Kulite type XCS-093-140mBARD) is placed in the wall right above theblade edge which we will refer too as ”top wall pressure transducer” (figure 4.12).The vortex shedding from the blade can be visualised by means of Schlieren tech-nique. The pressure can be measured at three positions by means of three miniaturepressure transducers (Kulite type XCS-093-140mBARD). The pressure transducersare placed in the edge 5mm from each other along the width of the edge model. Theslanted side of the edge model is a 0.4mm thick plate covering the pressure transduc-ers. Pressure holes of diameter 0.4mm are made in this plate in order to measure thepressure. The first hole is placed 2.0mm from the tip of the edge model, the secondone 4.0mm from the the tip and the third one 7.5mm from the tip in the middle ofthe slanted side. This is shown in figure 4.13. The pressures measured atthe bladeedge, will be compared with the pressures obtained in the numerical simulations.

In the Schlieren visualisation refractive index contrast is obtained by heating upthe edge with an infrared lamp (Philips Infraphil HP 3608) similar to the techniqueused by Disselhorst (1978). Before each flow visualisation the top plate of the set


l o u d s p e a k e r

b l a d e e d g e p r e s s u r e t r a n s d u c e r s

p r e s s u r e t r a n s d u c e r

w = 9 . 9 5 c m

1 9 . 8 c m

(a)

V 1

S

l o u d s p e a k e r 1 l o u d s p e a k e r 2

e d g e p r e s s u r e t r a n s d u c e r sp r e s s u r e t r a n s d u c e r

d =4 . 5 c m

V 2

p r e s s u r e t r a n s d u c e r s

t o p w a l lp r e s s u r e t r a n s d u c e r s

u ' 1 u ' 2

t b l a d e = 1 c m

h o t w i r e

2 0 . 9 c m 2 1 . 3 c m

1 9 . 8 c m

1 7 . 4 c m5 c m

d =4 . 5 c m

(b)

Figure 4.12: Drawing (a) shows a 3d picture of the set up and (b) a cross-section of the setup


2 2 3 . 5 5 5

p e d g e , 1 p e d g e , 2p e d g e , 3

(a)

t b l a d e = 1 0 . 0 3 . 0 m m

0 . 4 m m

0 . 4 m m

(b)

Figure 4.13: Drawing (a) shows a close-up of the pressure transducers in the edge shows(dimensions inmm) and (b) shows a cross-section of the edge model

up was removed to allow heating by means of the infra-red lamp. After heating theedge for a few minutes the set up was closed to allow experiments. A flash light,Flashpac 1100, is used as a light source and is triggered by a trigger unitgeneratinga TTL-pulse. The flash duration is typically 20µs. This signal is used to determinethe frequency of the input signal and delayed to obtain a stroboscopic effect. Pictureswere taken by a digital high speed camera, the Philips INCA 311, with accurateexternal triggering.

To calculate the velocity in the duct,u′, we assume a uniform pressure in volumeV1 andV2, so that:

Vidρ′

dt= −ρ0Su

′ − φloudspeaker , i = 1, 2 (4.11)

with Vi the volume in one of the two sections of the box,S = d × w the surfacearea of the connecting duct (see figure 4.12),ρ′ the density fluctuations,t the time,ρ0 = 1.2kg m−3 the ambient density andφloudspeaker is the mass displacement ofthe loudspeakers. Measurements have been carried out close to the resonance of theset up at frequencyf = 120 Hz. The quality factor determined by means of whitenoise excitation of the setup isq = 3. We neglectφloudspeaker. Usingρ′ = p′/c20,with c0 = 344m/s the speed of sound andp′ the pressure fluctuations, and assumingharmonic flow, we find:

u′ = −2iπfVip′

ρ0c20S. (4.12)

The velocity fluctuations,u′1, have also been measured using a hot wire anemome-ter (Dantec type 55P11 wire diameter 5µm with 55H20 support) placed 11.9cmfrom the top wall in the middle of the left duct. The hot wire measurements of the


velocity fluctuations agree within 25% with pressure fluctuation measurements whenusing equation 4.12. During other measurements the hot wire was removed. Theamplitude of the velocity fluctuations,uac = |u′|, are made dimensionless by meansof the Strouhal number, Srtblade

= ftblade/uac, with f the frequency andtblade thethickness of the blade. The time scale is related to the pressure in the reservoir V1,assuming it has acos (2πft) time dependence forsin (2πft) pressure fluctuations.

4.4.3 Measurements

Measurement have been obtained for acoustic flows withSrtblade= 0.2, 0.4, 0.8, 1.6

and 3.2. The flow separates at the tip of the edge generating vortices. In figure 4.14,some of the Schlieren pictures are shown. Att/T = 0 the flow starts flowing fromV1 to V2 (left to right) and att/T = 0.5 the flow changes direction and starts flowingfrom V2 to V1. The pictures, taken att/T = 0.3 and 0.8, give an impression of thesize of the vortices for Srtblade

= 0.2, 0.8, 3.2. For increasing Strouhal number thesize of the vortices decreases.

From the Schlieren visualizations, it is also observed that depending on theinitialconditions the flow can display two different modes (figure 4.15). In the first modethe first vortex is created above the slanted side of the edge model starting at t/T =0. A second vortex with a circulation of opposite sign is created when the flowchanges direction att/T = 0.5. Both vortices move as a vortex pair away from theedge (figure 4.15(a)). We will refer to this mode as ”mode 1”. In the second modethe first vortex is created starting att/T = 0.5 next to the edge model. A secondvortex of opposite sign is created starting att/T = 0 above the slated side of theedge. They move away from the edge as a vortex pair over the slanted side(figure4.15(b)). This mode will be referred to as ”mode 2”. While it is possible to forcethe flow in mode 2 (figure 4.15(b)), in most experiments mode 1 (figure 4.15(a)) isdominant. At Srtblade

= 1.6 the vortices that are created are small and it is no longerpossible to sustain mode 2 behaviour during a measurement. At even higher Strouhalnumber, Srtblade

= 3.2, the mode of vortex shedding changed spontaneously duringthe measurement from one mode to the other and back. These measurements are notincluded in this discussion.

The pressure is measured with the three pressure transducers at the edge, wherep′edge,1, p′edge,2 andp′edge,3 are the pressure transducer closest to the tip of the edge,the second closest and in the middle of the edge, respectively (figure 4.13), andp′top

the top wall pressure transducer right above the edge. The signal used to drive theloudspeaker is used to determine the period of oscillation. The pressure signals arephase averaged over6 × 103 periods. The absolute mean pressure that is establishedin the set up depends on leaks and is not reproducible. Because of the uncertain-ties in the mean pressure within the setup, the mean pressure measured at the top


(a) Srtblade=3.2

t/T=0.3(b) Srtblade

=3.2t/T=0.5

(c) Srtblade=3.2

t/T=0.8(d) Srtblade

=3.2t/T=1.0

(e) Srtblade=0.8

t/T=0.3(f) Srtblade

=0.8t/T=0.5

(g) Srtblade=0.8

t/T=0.8(h) Srtblade

=0.8t/T=1.0

(i) Srtblade=0.2

t/T=0.3(j) Srtblade

=0.2t/T=0.5

(k) Srtblade=0.2

t/T=0.8(l) Srtblade

=0.2t/T=1.0

Figure 4.14: Schlieren visualization of the flow for Srtblade= 3.2, 0.8 and0.2 at t/T = 0.3,

0.5, 0.8 and1.0.


(a) ”mode 1”

(b) ”mode 2”

Figure 4.15: Schlieren visualization of the two modes of vortex shedding: a) first vortex isformed on the right side of the edge, the vortex pair moves to the left away fromthe edge. b) first vortex is formed on the left, the vortex pairmoves to the right.

has been subtracted from the signal. Figure 4.16 shows the dimensionless pressure,p′/(ρ0c0uac), for mode 1 at Strouhal number Srtblade

= 0.2, 0.4, 0.8 and 1.6. As areminder small pictures in the plot illustrates the position of the vortices.

The pressure fluctuations at the top wall show that there is no perfect standingwave in the set up, because the pressure is not exactly in phase withp1. The samemeasurements are found by turning the top plate around180◦. Hence this is not due


0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 1 2

- 0 . 1

- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

t / T

p' edg

e,1/(r

c 0 u a

c)

S r = 0 . 2S r = 0 . 4S r = 0 . 8S r = 1 . 6

(a) transducer 1

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 1

- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

t / T

p' edg

e,2/(r

c 0 u a

c)S r = 0 . 2S r = 0 . 4S r = 0 . 8S r = 1 . 6

(b) transducer 2

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

t / T

p' edg

e,3/(r

c 0 u a

c)

S r = 0 . 2S r = 0 . 4S r = 0 . 8S r = 1 . 6

(c) transducer 3

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 1 5

- 0 . 0 1

- 0 . 0 0 5

0

0 . 0 0 5

0 . 0 1

0 . 0 1 5

t / T

p' top/(

r c 0

u ac)

S r = 0 . 2S r = 0 . 4S r = 0 . 8S r = 1 . 6

(d) top wall transducer

Figure 4.16: Dimensionless pressure fluctuations measuredfor four different Strouhal num-bers (Srtblade

= 0.2, 0.4, 0.8 and 1.6) at the pressure transducers in the edgemodel (a,b,c) for mode 1 vortex shedding and at the top of the set up (d).


to a misalignement of the top wall pressure transducer. This is not surprising, becauseof the asymmetry of the flow caused by the edge model, the absorbtion of sound bythe vortices and because of the leakage in the set up.

For0 < t/T < 0.4 the edge pressure is lower for Srtblade= 0.2 than for Srtblade

=1.6. This corresponds to the behaviour that is expected in a potential flow. Inorderunsteady potential flow the equation of Bernoulli reads:

ρ∂Φ

∂t+

1

2ρ|~v|2 + p = const(t) (4.13)

were the flow potential is given byΦ =∫

~v · d~x. At high Strouhal numbers, suchas Srtblade

= 1.6, the quadratic term1/2ρ|~v|2 is almost negligible and we observean almost harmonic pressure variation (linear behaviour). At low Strouhal numbers,such as Srtblade

= 0.2 , for 0 < t/T < 0.4 the non-linear term,1/2ρ|~v|2 induces adecrease of the local pressure, when there is only local flow separation. For0.7 <t/T < 1 we see from the flow visualization that there is a strong flow separationat the edge and the pressure trace indicates that the edge force has been suppressed.Considering the average pressure over a period of oscillation we see that the edgepressure is lower than the pressure at the top wall indicating an edge force.

Figure 4.17 show the dimensionless pressure,p′/(ρ0c0uac) measured at thesetwo transducers, for mode 2 behaviours at Srtblade

= 0.2, 0.4 and 0.8. As it was notpossible to sustain mode 2 vortex shedding behaviour for one measurement,mea-surements at Srtblade

= 1.6 are not included. We observe that on average the edgepressure is lower than the top-wall pressure indicating a net edge force.

4.4.4 Comparing measurements with results of the numerical simulation

To be able to compare the measurements with numerical simulations, the geometryof the duct of the set up is modelled. The geometry of the edge and the duct aroundthe edge is discretised using 2800 panels. Around the tip of the edge the density ofthe panels is increased. The calculations are carried out using 1000 equal time stepsper period. The desingularity parameter in the expressions for the velocityinducedby a vortex blob was chosen 10 times the time step.

Calculations with the blob method for the configuration considered fail to con-verge to a steady oscillatory solution if vorticity is not removed after some time.Vortex amalgamation methods are difficult to implement. We decided to use a verycrude approach. From the flow visualization it is observed that vortex shedding startsclose tot/T = 0. At that time the acoustical velocity is reversing from a flow fromV1 to volumeV2 towards a flow fromV2 to V1. After the reverse of the acousticalvelocity att/T = 0.5, a second vortex is shed containing opposite vorticity. The firstand the second vortex travel away as a vortex pair and seem to have little influence


0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 5

- 0 . 0 4

- 0 . 0 3

- 0 . 0 2

- 0 . 0 1

0

0 . 0 1

0 . 0 2

t / T

p' edg

e,1/(r

c 0 u a

c)

S r = 0 . 2S r = 0 . 4S r = 0 . 8

(a) transducer 1

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 6- 0 . 0 5- 0 . 0 4- 0 . 0 3- 0 . 0 2- 0 . 0 1

00 . 0 10 . 0 20 . 0 3

t / T

p' edg

e,2/(r

c 0 u a

c)

S r = 0 . 2S r = 0 . 4S r = 0 . 8

(b) transducer 2

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

t / T

p' edg

e,3/(r

c 0 u a

c)

S r = 0 . 2S r = 0 . 4S r = 0 . 8

(c) transducer 3

0 0 . 2 0 . 4 0 . 6 0 . 8 1- 0 . 0 1 5

- 0 . 0 1

- 0 . 0 0 5

0

0 . 0 0 5

0 . 0 1

0 . 0 1 5

t / T

p' top/(

r c0 u

ac)

S r = 0 . 2S r = 0 . 4S r = 0 . 8

(d) top wall transducer

Figure 4.17: Dimensionless pressure fluctuations measuredfor three different Strouhal num-bers (Srtblade

= 0.2, 0.4 and 0.8) at the pressure transducers in the edge model(a,b,c) for mode 2 vortex shedding and at the top of the set up (d).


on the next vortex shedding. This allows to use the ”vortex-blob” method, during onesingle oscillation period, starting without vortices.

Figure 4.18 shows the vortex distribution computed by the vortex-blob method.Figure 4.14 and figure 4.18 show a good resemblance.

To compare the pressure measured in the set up and the pressures calculated usingthe vortex blob method, we calculated the difference between the pressureat thethree locations on the slanted side of the edge model and the pressure at thetop wall.The results are found in figure 4.19 for all three pressure transducers in the edge asfunction of time. The left side shows the calculations using the blob method and onthe right side the measured pressures are shown.

The measurements and the numerical simulation show similarities in shape, how-ever the fluctuations in the pressure obtained for the vortex blob method arelargerand the peaks are less wide.

A comparison is also done for mode 2 vortex shedding. For these calculationst/T = 0 is defined as the start of the flow going fromV2 to V1, the flow startswith the single vortex left of the edge. The difference between the pressures on theedge at the three locations and the top pressure right above the edge is plotted forSrtblade

= 0.2, 0.4 and 0.8 in figure 4.20.Although, we find some similarities in the variation with time, the effect of the

vortex pair travelling away from the edge has a large effect on the calculations inmode 2. From the prediction of vorticity distribution using the vortex blob method,we can see that while the vortex pair moves away a part of the vortex remainsclose tothe edge. However, the visualisations do not show this vortex left behind.The sameeffect takes place for mode 1 behaviour (figure 4.18(c,d,g,h and k)). This vortex hasless effect on the edge pressure, because it is not close to the edge asin the case ofmode 2.

4.5 Conclusions

Representing the rotor blade by a flat plate and the flow separation at the sharp edgeof the blade by a point vortex, a model is obtained allowing to predict the drivingtorque on a rotor placed in an oscillatory flow. A simplified model is proposed forhigh Strouhal number (SrLblade

>> 1) which provides an explicit algebraic expres-sion without the need to determine the details of the flow. Comparison between thetwo models indicates that they are equivalent within the accuracy of the performedexperiments. The results show that the thickness of the plate is an important fac-tor for occurrence of spurious counts. The presence of a thick trailingedge in tur-bine blades increases the critical acoustical pulsation amplitude above whichspuriouscounts appear. The model provides a prediction of the order of magnitudeof the crit-ical torque, and can be used to determine typical conditions for the occurrence of

4.5. Conclusions 105

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.3

(a) Srtblade=3.2

t/T=0.3

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.5

(b) Srtblade=3.2

t/T=0.5

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.8

(c) Srtblade=3.2

t/T=0.8

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 1

(d) Srtblade=3.2

t/T=1.0

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.3

(e) Srtblade=0.8

t/T=0.3

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.5

(f) Srtblade=0.8

t/T=0.5

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.8

(g) Srtblade=0.8

t/T=0.8

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 1

(h) Srtblade=0.8

t/T=1.0

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.3

(i) Srtblade=0.2

t/T=0.3

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.5

(j) Srtblade=0.2

t/T=0.5

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 0.8

(k) Srtblade=0.2

t/T=0.8

−1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3 t/T = 1

(l) Srtblade=0.2

t/T=1.0

Figure 4.18: Prediction of the vortex distribution of the flow for Srtblade= 3.2, 0.8 and0.2

at t/T = 0.3, 0.5, 0.8 and1.0.


0 0.2 0.4 0.6 0.8 1−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

t/T

(ped

ge,3

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(a) blob method,pedge,1

0 0.2 0.4 0.6 0.8 1−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

t/T

(ped

ge,3

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(b) measurement,pedge,1

0 0.2 0.4 0.6 0.8 1

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

t/T

(ped

ge,2

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(c) blob method,pedge,2

0 0.2 0.4 0.6 0.8 1

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

t/T

(ped

ge,2

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(d) measurement,pedge,2

0 0.2 0.4 0.6 0.8 1−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

t/T

(ped

ge,1

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(e) blob method,pedge,3

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

t/T

(ped

ge,1

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8Sr=1.6

(f) measurement,pedge,3

Figure 4.19: The dimensionless difference between pressure pedge,1, pedge,2 andpedge,3 andthe top wall right above the edge for Srtblade

= 0.2, 0.4, 0.8 and 1.6. Themode of vortex shedding is mode 2. On the left side the pressure difference aspredicted using the vortex blob method. On the right side themeasured pressuredifference is shown.

4.5. Conclusions 107

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

t/T

(ped

ge,3

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8

(a) blob method,pedge,1

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

t/T

(ped

ge,1

−p to

p)/(ρ

c0 u

ac )

Sr=0.2Sr=0.4Sr=0.8

(b) measurement,pedge,1

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

(ped

ge,2

−p to

p)/(ρ

c0 u

ac)

t/T

Sr=0.2Sr=0.4Sr=0.8

(c) blob method,pedge,2

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

t/T

(ped

ge,2

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8

(d) measurement,pedge,2

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t/T

(ped

ge,1

−p to

p)/(

ρ c 0 u

ac)

Sr=0.2Sr=0.4Sr=0.8

(e) blob method,pedge,3

0 0.2 0.4 0.6 0.8 1−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t/T

(ped

ge,3

−p to

p)/(ρ

c0 u

ac)

Sr=0.2Sr=0.4Sr=0.8

(f) measurement,pedge,3

Figure 4.20: The dimensionless difference between the pressurepedge,1, pedge,2 andpedge,3

and the top wall right above the edge for Srtblade0.2, 0.4 and 0.8. The mode of

vortex shedding is mode 2. On the left side the pressure difference as predictedusing the vortex blob method. On the right side the measured pressure differenceis shown.


spurious counts in field conditions. In view of its simplicity, the theory for the limitcase,SrLblade

>> 1, is a useful engineering tool in the prediction of the occurrenceof spurious counts.

A model describing details of the flow around the sharp trailing edge of a rotorblade is proposed. This model is based on the vortex blob method. In this model weassume that vortex shedding is initiated at the maximum of the pressure (t/T = 0)and that the vortex shed in the second half of a period annihilates the first vortex afterone oscillation period (t > T ). The predicted vortex structure agrees qualitativelywith the flow visualization and the model predicts reasonably well the dependenceof the edge pressure on the Strouhal number. However, to obtain a prediction of theedge force a more accurate numerical model is necessary.

5

Conclusions

5.1 Introduction

When used at ideal conditions, gas turbine flow meters allow measurement ofthevolume flow with an accuracy of about 0.2%. During use systematic changesinresponse can be induced by wear or damage of the rotor. Systematic errors can alsobe induced by perturbations in the flow (non-uniformities, swirl or pulsations). Thisstudy is limited to the behaviour of turbine flow meters for three different typesofflow:

• steady flow (Chapter 2)

• main flow with acoustic pulsations (Chapter 3)

• acoustic pulsation without main flow (Chapter 4)

We exclude the swirl and other vortical perturbations of the flow. The acousticalpulsations induce a fluctuation in velocity which is uniform across the rotor. In allthese cases the behaviour of the turbine meter is analysed by comparison ofthe re-sults of analytical models with the results of experiments. Some of these simplifiedmodels can be used to apply corrections or as support for design rules inengineeringapplications.

5.2 Stationary flow

An ideal helicoidal turbine meter without mechanical friction nor fluid drag willhavea rotational speed,ω, proportional to the volume flow,Q: ω = KQ. In practicethe meter constantK determined by calibration will display a slight dependence on

110 5. Conclusions

Reynolds number. The aim of the designer is to obtain a meter with an almost con-stantK on one side but also to achieve a rotor which is robust. This means thatK isnot strongly affected by wear and other damage of the meter. When a rotorwith highsolidity is used the flow leaves the rotor in the directions tangentially to the blades.We call this ”full fluid guidance”. We expect that such rotors are less sensitive towear or impact than rotors with lower solidity. We therefore focus on rotorswithhigh solidity.

Results of a two-dimensional model described in chapter 2 have been comparedto the calibration data of Elster-Instromet for two different turbine flow metergeome-tries. The theory assumes that the flow is similar to that in a two-dimensional cascadewith full fluid guidance. The theory predicts global trends. Small changesin rotorblade geometry cannot be explained by the model (section 2.6.1).

It is clear from measurements that the velocity profile at the inlet of the rotor isReynolds number dependent (section 2.3). This appears to have a significant effecton the deviation,E = ω−ωid

ωid, of the rotation from the ideal behaviour of a helicoidal

rotor.Wind tunnel measurements on a model of a rotor blade provide an indication for

the influence of the shape of the rotor blades on the drag of the rotor (section 2.4.1).These measurements also show that for a typical rotor blade profile zero angle

of incidence with respect to the flow,α = 0, there is a lift force induced by theasymmetric shape of the trailing edge of the blade. This partially explains why theactual rotation speed is higher than the ideal rotation speed,E > 0.

It is found that the geometry of the tip clearance between the blades and thepipe wall has an important effect on the deviations from ideal flow (section2.6.3).By placing the tip of the rotor blade in a cavity in the pipe wall the manufacturerscan obtain a rotor constantK which is almost independent of the Reynolds number.A theoretical model of the tip clearance flow effect should therefore be included toobtain a better prediction of the Reynolds number dependence of the flow meter.

5.3 Main flow with pulsations

In chapter 3 pulsating flows are investigated for high pulsation frequency. At thesehigh frequencies the inertia of the rotor is so important that a steady rotation speed ofthe rotor is achieved , so thatdω

dt= 0. Assuming a quasi-steady behaviour of the flow

the model obtained for steady flow ( chapter 2) can be used to predict the effect ofpulsations. This quasi-steady flow model predicts that the deviationEpuls = ω−ω0

ω0

from the steady flow response depends quadratically on the r.m.s value of the relative

velocity amplitudeEpuls =

(

|u′

in|2

u2in

)

, is compared with measurements, whereuin is

the velocity at the rotor inlet. While significant departure from this model are found,

5.4. Pulsations without main flow 111

the quadratic dependence of the relative measurement error remains. Wetherefore

introduce a quadratic fit parametera such thatEpuls = a

(

|u′

in|2

u2in

)

. The error caused

by the pulsation is dependent on Strouhal number (section 3.6.1). The quadratic fit

parameter,a, of the measurement errors decreases as Sr1

5

Lbladefor SrLblade

< 2.5. Thisparticular power law is not yet understood. This quadratic fit parameter isnot depen-dent on Reynolds number (section 3.6.2). Measurements at higher Strouhal numberwere not accurate enough to confirm this dependence at higher Strouhal numbers.The deviation obtained for the superposition of two harmonic perturbations can bepredicted by addition of the deviations caused by the two individual perturbations.

Tests with a different rotor indicate that the blade shape does not affectthe pul-sation error,Epuls (section 3.6.4).

Determination of the velocity amplitude at the rotor is critical for correcting themeasurements. Local measurements of the velocity either by hot wire anemometersor pressure transducers are not reliable (section 3.4.4). More globalmeasurementsusing multiple pressure transducers in a microphone array set up are necessary.

5.4 Pulsations without main flow

The behaviour of turbine meters in a purely oscillatory flow was studied experimen-tally in chapter 4, with a simplified theoretical model and a discrete vortex model.Pulsations without main flow can induce rotation of rotors with blades with a roundedleading edge and a sharp trailing edge. Experiments using a rotor with sharptrail-ing and leading edges show that no ghost counts occur, even when the symmetry isbroken by an initial rotation of the rotor (section 4.3.2).

An explicit equation calculating the edge force on an isolated flat plate usingpotential flow theory yields an engineering tool to predict the onset of ghost counts(section 4.2.4). This equation can be easily extended to a cascade of blades.

Experiments were carried out to obtain more insight into the influence of thethickness of the rotor blade on the edge force. A comparison of the visualisation ofthe flow around the edge of a blade with the discrete vortex potential flow solutionshows reasonable agreement. Similar characteristics in the pressure at theedge arefound in the results of the discrete vortex model and the experiments (section4.4.4).

5.5 Recommendations

The aim of this study was to develop and verify simplified analytical models to beused in predicting the behaviour of turbine flow meters at different flow conditions.

For steady flow, the two dimensional model is able to explain global effects on

112 5. Conclusions

the deviations from real flow. For a designer the model can identify sensitive pointsin the design, but a better prediction of the Reynolds number dependence can only beachieved by including realistic tip clearance effects in the theory.

The quasi-steady flow model predicting the measurement error during pulsatingflow provides a reasonable (within 40%) prediction for Strouhal numbersbased onthe blade tip chord length, SrLblade

up to 10. For Strouhal number pulsations upto 2.5 an empirical relation (equation 3.26) can be used to correct the quasi-steadytheory for a more accurate prediction. It is recommended to extend this equation tohigher Strouhal numbers. This would involve new experiments with more accurateacoustical velocity measurements at high frequencies.

Problem in the use of this model in practice is determining the velocity amplitudeof the pulsations at the rotor. Placing an array of microphones within the flowmeterappears to be the most promising option.

Using blades with a rounded leading edge and sharp trailing edge enhances thesteady performance of a rotor, but can result in spurious counts in purely oscillatingflow. An analytical expression derived in chapter 4 gives an order ofmagnitudeprediction of the onset of spurious counts in purely oscillating flow (withoutmainflow). Using this tool possible problems of ghost counts can be located. Byusing thecascade theory of Durant (1963) this model can be extended to include interactionbetween blades. Furthermore, a more accurate determination of the edge force usinga numerical method can improve the accuracy of the prediction.

APPENDIX

A

Mach number effect in temperaturemeasurements

Assuming that the temperature sensor placed in the meter measures a temperatureclose to the adiabetic wall temperature,Tw, for a turbulent boundary layer on a flatplate we have (Shapiro, 1953):

Tw

T∞≈ 1 +

√Prγ − 1

2M2 , (A.1)

where Pr is the Prandtl number,γ is Poisson’s ratio andT∞ is the main flow temper-ature. In general the meter is calibrated against a series of three other meters placedin parallel, so that the Mach number at the meter being calibrated is about a factor 3higher than at the reference meters. AssumingTw ≈ T∞ induces a calibration errorin air of the order0.18M2.

116 A. Mach number effect in temperature measurements

B

Boundary layer theory

B.1 Introduction

This appendix explains some basic concepts of boundary layer theory. It is by nomeans a complete analysis, only some simplified boundary layer calculations used inthis thesis will be addressed.

The flow in the turbine flow meters investigated in this thesis operate at flowconditions with high Reynolds numbers, Re= UL

ν, with U the main stream velocity,

L a characteristic length scale andν the kinematic viscosity. For high Reynoldsnumber flows, Re>> 1, viscous forces can be neglected, except within a thin layernear the wall. Near the wall viscous forces are dominant, causing the fluid tostickto the wall. This thin layer can be described by boundary layer theory, whilefor thebulk of the flow an inviscid flow method, such as one based on the Euler equations,can be used. An example of a boundary layer is shown in figure B.1.

yd ( x )u ( x , y )

x

U

U

Figure B.1: Boundary layer on a flat plate in parallel flow

In the boundary layer the mass conservation and Navier-Stokes equation(equa-

118 B. Boundary layer theory

tions 2.1 and 2.2) can be reduced to,

∂u

∂x+∂v

∂y= 0 ,

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ ν

∂2u

∂y2,

− 1

ρ

∂p

∂y= 0 ,

(B.1)

where thex-axis is along the wall and they-axis is perpendicular to the wall,u isthe velocity in thex-direction andv is the velocity in they-direction. The pressurevariation in streamwise direction in the bulk flow is imposed on the boundary layerand can be found using the Euler equation for the bulk flow. For the steadyflow alongthe flat plate this leads to;

∂p

∂x= −ρU(x)

∂U(x)

∂x(B.2)

The thickness of this boundary layer,δ, is difficult to determine, because the ve-locity increases smoothly from zero at the wall and reaches asymptotically to thefree stream velocity,U , giving it no exact limit. A well-defined quantity is the dis-placement thickness,δ1. The displacement thickness is the distance at which a solidboundary would be placed in order to keep the mass flux equal to the mass flux of theflow with boundary layers;

δ1(x) =

∫ δ

0

(

1 − u

U

)

dy =

∫ ∞

0

(

1 − u

U

)

dy , (B.3)

where the upper limit of the integrant is extended to infinity, because fory ≥ δ u = Uand the integrand is zero.

Another useful and well-defined quantity is the momentum thickness,δ2. Themomentum thickness is defined to account for the loss of momentum for the casewhere the flow has no boundary layer, but is corrected with the displacement thick-ness and the actual flow;

δ2(x) =

∫ ∞

0

u

U

(

1 − u

U

)

dy . (B.4)

For a uniform flow through a channel of height,H, the total momentum flux is equalto ρU2 (H − 2 (δ1 + δ2)).

For some simple cases an exact solution of the boundary layer equations B.1canbe found, for example the Blasius solution for flat plates (discussed in section B.2),but for most flow problems there are no exact solutions and approximate ornumericalmethods have to be used. An example of an approximate method is the Von Karmanintegral momentum equation, this equation will be discussed in section B.3.

B.2. Blasius exact solution for boundary layer on a flat plate 119

B.2 Blasius exact solution for boundary layer on a flat plate

Blasius considered the boundary layer along a semi-infinite flat plate (Schlichting,1979) (see figure B.1). The flow has a constant, steady, velocity,U , parallel to thex-axis and there is no pressure gradient. The boundary layer equationsB.1 reduce to:

∂u

∂x+∂v

∂y= 0 ,

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2.

The boundary condition are given by the no-slip condition at the wall, aty = 0u = v = 0, and the condition of smooth transition from the boundary layer to themain stream velocity, aty = ∞ u = U . Blasius supposed that the dimensionlessvelocity, u

U, at various distances from the edge is self-similar, i.e. depends ony/δ(x),

making it possible to make the variables non-dimensional, using the boundary layerthickness,δ(x) and the main stream velocity,U . The boundary layer thickness basedon the solution for a suddenly accelerated plate as derived by Stokes (Schlichting,1979) is found to be of the form;δ ∼ νx

U. Usingη ∼ y

δ

η = y

√

U

νx, (B.5)

and introducing a stream function,ψ;

ψ =√νxUf(η) , (B.6)

the second-order partial differential equation can be transformed in a third-order or-dinary differential equation;

2d3f

dη3+ f

d2f

dη2= 0 , (B.7)

with boundary conditions;

η = 0 f = 0 anddfdη

= 0 ,

η → ∞ dfdη

= 1 .

More details can be found in Schlichting (1979). To solve this equation analyticallyis difficult and was done by Blasius using power series expansion.


Howarth (1938) solved this equation numerically. With the velocity profile known,the shear stress,τw, on the plate caused by the viscous flow can be found as:

τw(x) = µ

(

∂u

∂y

)

y=0

= ρU

√

Uν

xf ′′(0) ≈ 0.332ρU

√

νU

x. (B.8)

Using equation B.3 the displacement thickness of a boundary layer on a flatplatebecomes;

δ1(x) ≈ 1.7208

√

νx

U. (B.9)

B.3 The Von Karman integral momentum equation

As mentioned above for most problems an exact solution cannot be found.For theseproblems an approximate method can be used. Integrating the momentum equationfor steady flow, a global solution can be found;

∫ h

y=0

(

u∂u

∂x+ v

∂u

∂y− U

dUdx

)

dy =

∫ h

y=0ν∂2u

∂y2dy . (B.10)

whereh(x) is outside the boundary layer for all values ofx.Using the definition of shear stress,τw;

τw = µ∂u

∂y|y=0 , (B.11)

and replacing the normal velocity component,v, using the continuity equation with;

v = −∫ y

0

∂u

∂xdy , (B.12)

equation B.10 becomes:

∫ h

y=0

(

u∂u

∂x− ∂u

∂y

∫ y

0

∂u

∂xdy − U

dUdx

)

dy = −τwρ. (B.13)

By integrating the second term on the left-hand side by parts, the equation can berewritten as;

∫ h

y=0

∂

∂x(u (U − u)) dy +

dUdx

∫ h

y=0(U − u) dy =

τwρ. (B.14)

B.4. Description laminar boundary layer 121

Using the definitions of displacement thickness,δ1, (equation B.3) and momen-tum thickness,δ2, (equation B.4) we get the Von Karman equation:

ddx

(

U2δ2)

+ δ1UdUdx

=τwρ. (B.15)

To solve this equation a velocity profile,uU

, has to be assumed. With this veloc-ity profile the displacement thickness, the momentum thickness and the shear stressat the wall can be determined and the Von Karman equation can be solved. Thenext sections will introduce an approximate laminair and turbulent boundarylayerdescriptions used in this thesis.

B.4 Description laminar boundary layer

Pohlhausen (1921) used a fourth-order polynomial to describe a laminarboundarylayer, however, Hofmans (1998) and Pelorson et al. (1994) found that a third-orderpolynomial gives a more accurate description of the boundary layer. Thevelocity inthe boundary layer can than be described as:

u

U=

3∑

i=0

ai

(y

δ

)i

. (B.16)

To satisfy the no-slip condition at the wall and a smooth transition to the mainstream velocity at the edge of the boundary layer the following four boundary condi-tions are introduced;

for y = 0 : u = 0 ν∂2u

∂y2= −U dU

dx,

for y = δ : u = U∂u

∂y= 0 .

Introducing the non-dimensional parameter quantity,λ = δ2

νdUdx

, the coefficientsof the polynomial can be found;

a0 = 0 ,

a1 =3

2+λ

4,

a2 = −λ2,

a3 = −1

2+λ

4.


Using the definitions for the displacement thickness,δ1, (equation B.3) and themomentum thickness,δ2, (equation B.4), it can be found that;

δ1 =

(

3

8− 1

48λ

)

δ ,

δ2 =

(

39

280− 1

560λ− 1

1680λ3

)

δ .

If the pressure gradient is neglected,λ = 0, the exact solution of Blasius can befound.

Using this third-order polynomial description the shear stress,τw, at the wall canbe calculated and is:

τw = µ

(

∂u

∂y

)

y=0

= a1µU

δ. (B.17)

B.5 Description turbulent boundary layer

If fully turbulent flow is considered the velocity and pressure componentscan beseparated into a mean motion and a fluctuation,

u = u+ u′ v = v + v′ p = p+ p′ . (B.18)

To approximate the turbulent boundary layer, we are only interested in averagevelocities and one has to realize that there are no exact solutions for turbulent flow.Using Prandtl’s mixing-length theory (Schlichting, 1979) it can be calculated, thatthe velocity profile for a turbulent flow is quite complex composition of logarithmicfunction and an additional linear layer. However, for many practical applications, ithas been shown experimentally that the velocity profile can be approximated by asimple equation in the form of a 1/7th power (Schlichting, 1979);

u

U=(y

δ

)1

7

. (B.19)

Using this equation the displacement thickness,δ1, and the momentum thickness,δ2are;

δ1 =1

8δ ,

δ2 =7

72δ .

Although this velocity profile is a good approximation, it has some shortcomings.A problem with this profile is that at the wall the gradient of the velocity becomes

B.5. Description turbulent boundary layer 123

infinite and the transition to the main flow. As a consequence of this it is impossibleto calculate the shear stress caused by the boundary layer on the wall anda empiricalrelation has to be found. In this thesis, as an approximation the shear stressfound fora fully developed turbulent pipe flow;

τw = 0.0225ρU2( ν

Uδ

)1

4

(B.20)


C

Measurements

C.1 Introduction

The effect of velocity pulsations on flow measurements has been investigated in thisthesis by measuring this error at resonance frequencies between 24Hz and 730Hzand relative velocity amplitudes,uac/u0, ranging from about 0.01 to 2. In this ap-pendix the result are represented for the measurements carried out at pulsation fre-quencies of 24, 117, 360 and 164Hz. The data is plotted for every pulsation fre-quency separately using a double logarithmic scale as well as a linear scale.Themainstream velocity at which the measurement is performed are indicated by thedifferent symbols.

126 C. Measurements

C.2 Pulsation frequency of 24Hz

10−2

10−1

100

10−4

10−3

10−2

10−1

100


rela

tive

mea

sure

men

t err

or, E

puls


0 = 10 m/s

u0 = 2 m/s

u0 = 5 m/s

Figure C.1: The relative measurement error,Epuls, as a function of the relative amplitudeof the pulsations,|u′|/u0, for measurements at a pulsation frequency of 24Hz.Plotted using double logarithmic scale

0 0.5 1 1.5 20

0.5

1

1.5

2


rela

tive

mea

sure

men

t err

or, E

puls


0 = 10 m/s

u0 = 2 m/s

u0 = 5 m/s

Figure C.2: The relative measurement error,Epuls, as a function of the relative amplitude ofthe pulsations,|u′|/u0, for measurements at a pulsation frequency of 24Hz.

C.3. Pulsation frequency of 69Hz 127


10−2

10−1

100

10−4

10−3

10−2

10−1

100


rela

tive

mea

sure

men

t err

or, E

puls


0 = 5 m/s

u0 = 0.5 m/s

u0 = 1 m/s

u0 = 1.5 m/s

u0 = 2 m/s

u0 = 3 m/s

Figure C.3: The relative measurement error,Epuls, as a function of the relative amplitudeof the pulsations,|u′|/u0, for measurements at a pulsation frequency of 69Hz.Plotted using double logarithmic scale

0 0.5 1 1.5 20

0.5

1

1.5

2


rela

tive

mea

sure

men

t err

or, E

puls


0 = 5 m/s

u0 = 0.5 m/s

u0 = 1 m/s

u0 = 1.5 m/s

u0 = 2 m/s

u0 = 3 m/s


128 C. Measurements


10−2

10−1

100

10−4

10−3

10−2

10−1


rela

tive

mea

sure

men

t err

or, E

puls


0 = 3 m/s

u0 = 1 m/s

u0 = 2 m/s

Figure C.5: The relative measurement error,Epuls, as a function of the relative amplitude ofthe pulsations,|u′|/u0, for measurements at a pulsation frequency of 117Hz.Plotted using double logarithmic scale

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5


rela

tive

mea

sure

men

t err

or, E

puls


0 = 3 m/s

u0 = 1 m/s

u0 = 2 m/s


C.5. Pulsation frequency of 363Hz 129


10−2

10−1

100

10−4

10−3

10−2

10−1

100


rela

tive

mea

sure

men

t err

or, E

puls


0 = 15 m/s

u0 = 1 m/s

u0 = 2 m/s

u0 = 5 m/s

u0 = 10 m/s


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


rela

tive

mea

sure

men

t err

or, E

puls


0 = 15 m/s

u0 = 1 m/s

u0 = 2 m/s

u0 = 5 m/s

u0 = 10 m/s


130 C. Measurements


10−2

10−1

100

10−4

10−3

10−2

10−1


rela

tive

mea

sure

men

t err

or, E

puls


0 = 15 m/s

u0 = 1 m/s

u0 =2 m/s

u0 = 5 m/s

u0 = 10 m/s


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


rela

tive

mea

sure

men

t err

or, E

puls


0 = 15 m/s

u0 = 1 m/s

u0 =2 m/s

u0 = 5 m/s

u0 = 10 m/s

Figure C.10: The relative measurement error,Epuls, as a function of the relative amplitudeof the pulsations,|u′|/u0, for measurements at a pulsation frequency of 730Hz.

D

Force on leading edge

Using the transformation of Joukowski (see equation 4.1), the value of dξ/dz can becalculated close to the leading (singular) edge.

limz→−2A

[

dξdz

]

= limz→−2A

[

1

2+

z

2√

(z − 2A)(z + 2A)

]

= limz→−2A

i√A

2√z + 2A

(D.1)

where A is the radius of the circle in theξ-plane. Because near the edge (z →−2A) dξ

dzbecomes very large and the second term becomes the dominant term and

equation D.1 is obtained. The potential of the flow is given by equation 4.2. Withthisequation and the Kutta condition imposed at the trailing edge,z = 2A, (see equation4.3) the circulation,Γv, can be found:

Γv = 4πuac sinαA(A− ξv)(ξ

∗v −A)

ξvξ∗v −A2(D.2)

whereuac is the acoustic oscillation amplitude,α is the incidence of the flow andξv is the position of the vortex in the transformed plane. Using the flow potential,Φand the circulationΓv, dΦ/dξ close to the leading edge can be calculated:

[

dΦ

dξ

]

ξ→−A

= uace−iα − uace

iα − iΓv

2π

(

1

−A− ξv+

ξ∗v−Aξ∗v −A2

)

= −2iuac sinα

(

1 +(A− ξv)(A− ξ∗v)

(A+ ξv)(A+ ξ∗v)

)

(D.3)

By combining equations D.1 and D.3 it follows that:

132 D. Force on leading edge

[

dΦ

dz

]

z→−2A

=

[

dΦ

dξ

]

ξ→−A

[

dξdz

]

z→−2A

= uac

√A sinα

(

1 +(A− ξv)(A− ξ∗v)

(A+ ξv)(A+ ξ∗v)

)

1√z + 2A

(D.4)

The force on the edge can be found by utilising Blasius’ theorem for the force,i.e. evaluating an integral around the closed contourǫ

Fx − iFy =iρ

2limǫ→0

∮

ǫ

(

dΦ(z)

dz

)2

dz (D.5)

whereFx denotes the force parallel to the plate (the edge force) andFy representsthe force perpendicular to the plate.

Using the Cauchy integral theorem, the force becomes:

Fx = Fe = −πρu2acA sin2 α

(

1 +(A− ξv)(A− ξ∗v)

(A+ ξv)(A+ ξ∗v)

)2

= −4πρu2acA sin2 α

(

A2 + ξvξ∗v

(A+ ξv)(A+ ξ∗v)

)2

(D.6)

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Summary

Dynamics of turbine flow meters

Axial turbine flow meters are used in applications, in which accurate volume flowmeasurements are desired. When used at ideal conditions, turbine flow meters allowgas flow measurements with an accuracy of about 0.2%. The aim of our research is todevelop engineering tools allowing to design robust and accurate turbine flow meters.

Ideally, the volume flow is proportional to the rotation speed. However, slightdeviations are observed as a function of the Reynolds number. In designing a flowmeter, the object is to make the dependence on the Reynolds number as small aspos-sible and to create a robust meter, not sensitive to wear nor damage. In thisthesis wedescribe a two-dimensional analytical model to predict the deviation of the rotationspeed of the rotor from the rotation speed of an ideal helical rotor in an ideal flow,without mechanical friction or fluid drag. By comparing the results of the model withcalibration data of turbine flow meters provided by Elster-Instromet, we find that thetheory can explain global effects of the inlet velocity profile, the pressure drag ofthe wake and other friction forces. For a more accurate model it is necessary to in-clude realistic tip clearance effects and the additional lift induced by the shape of thetrailing edge.

Another important source of systematic errors are time dependent perturbationsin the flow. We investigate the effect of pulsations at high frequencies on the rota-tion speed of the rotor. Above a critical frequency, determined by the inertia of therotor, the turbine meter will not be able to follow the variations in volume flow intime. Instead an average rotation speed will be established. Due to non-linearities ofthe forces exerted by the flow on the turbine blades this rotation speed correspondsto a higher steady volume flow than the actual time-averaged flow. By assumingquasi-steady behaviour of the flow for the model obtained for ideal steady flow de-scribed above, the relative error in volume flow is equal to the root-mean-square ofthe ratio of acoustical to main flow velocities. The range of validity of this predic-

138 Summary

tion has been explored experimentally for harmonic pulsations. Although significantdeviations from the quasi-steady model were found, the quadratic dependence on thevelocity amplitude appears to remain valid for all measurements. The exact quadraticdependence is a function of the Strouhal number of the pulsations. In the range ofStrouhal numbers below 2.5, based on the blade chord length at the tip of therotorblade and the flow velocity at the rotor inlet plane, we find a slow decrease of theerror with increasing Strouhal number following a power1

5 of Strouhal. Measure-ments at high Strouhal numbers were not reliable enough to confirm this dependencefor higher Strouhal numbers. The deviation obtained for the superposition of twoharmonic perturbations can be predicted by addition of the deviations caused by thetwo individual perturbations.

For the extreme case in which we have pulsations but no time-averaged main flow,it is possible that the rotor starts rotating above a critical amplitude of the pulsations.These surious counts or ”ghost counts” are caused by the shape of the rotor blades.Using potential flow theory for thin rotor blades an explicit equation is developed topredict the onset of the ghost counts. This equation is verified by experiments to be anengineering tool allowing to predict the order of magnitude of the onset of spuriouscounts. A more accurate prediction of the onset of spurious counts can be found bystudying the flow around the rotor blades. Experiments were performed to measurethe pressure at three different locations on a scale model of the edge ofa rotor bladeand to visualise the flow around this edge. These measurements and visualisationsshow reasonable agreement with the results of a discrete vortex blob model.

Samenvatting

Dynamica van turbinedebietmeters

Wanneer het nodig is om het volumedebiet nauwkeurig te meten, worden vaakaxiale turbinedebietmeters toegepast. Onder ideale omstandigheden kunnenturbine-debietmeters voor gasstromingen een nauwkeurigheid bereiken tot 0.2%. Het doelvan dit onderzoek is om technologische applicaties te ontwikkelen die het mogelijkmaken robuuste en nauwkeurige meters te ontwerpen.

In het ideale geval is het volumedebiet rechtevenredig met de rotatiesnelheidvan de rotor. In praktijk echter treden er kleine afwijkingen op als functie vanhet Reynoldsgetal. Bij het ontwerpen van een turbinemeter is het doel omdezeafhankelijkheid van het Reynoldsgetal zo klein mogelijk te maken. Tegelijkertijdmoet de turbinemeter ook robuust zijn, zodat hij niet gevoelig is voor slijtageenkleine beschadigingen. In dit proefschrift wordt een twee-dimensionaal analytischmodel beschreven dat het mogelijk maakt afwijkingen te voorspellen van derotatie-snelheid van de rotor met de rotatiesnelheid van een ideale rotor, waarbij de bladende vorm hebben van een ideale helix, die ronddraait zonder mechanische wrijvingof stromingswrijving. De resultaten van dit model zijn vergeleken met ijkmetingendie beschikbaar zijn gesteld door Elster-Instromet. Uit deze vergelijking kunnen weconcluderen dat het model de globale effecten van het instroomprofiel,van de druk-weerstand van het zog en van andere wrijvingskrachten verklaart. Omde invloedvan de ruimte tussen de uiteinden van de rotorbladen en de wand te verklaren is eenuitgebreider model nodig. Dit geldt ook voor het effect van de additionele liftkrachtdie opgewekt wordt door de vorm van de achterkant van de rotorbladen.

Een andere belangrijke bron van systematische fouten zijn tijdsafhankelijkever-storingen in de stroming. Het effect van pulsaties met hoge frequenties opde rotatie-snelheid van de rotor is onderzocht. Boven een kritische frequentie, diebepaald wordtdoor de massatraagheid van de rotor, is de rotor niet meer in staat de variaties in hetvolumedebiet te volgen. In plaats hiervan stelt zich een gemiddelde rotatiesnelheid

140 Samenvatting

in. Door niet-lineariteit van de krachten die door de stroming worden uitgeoefendop de turbinebladen, correspondeert deze rotatiesnelheid met een volume debiet datgroter is dan het tijdsgemiddelde debiet. Door aan te nemen dat het gedrag van destroming quasi-stationair is en door gebruik te maken van het model dat hierbovenis beschreven, vinden we dat de fout in het gemeten volumedebiet gelijk is aan derms-waarde van de verhouding tussen de akoestische snelheid en de hoofdstroom-snelheid. We hebben de grenzen onderzocht waarbinnen dit model voor harmonischepulsaties geldig is. Alhoewel we significante afwijkingen van de quasi-stationairetheorie hebben gevonden, lijkt de kwadratische afhankelijkheid van de snelheidsver-houding geldig te blijven. De exacte kwadratische afhankelijkheid is een functie vanhet Strouhalgetal. Voor Strouhalgetallen, gebaseerd op de koorde vande bladen aande uiteinden en de snelheid bij het binnengaan van de rotor, kleiner dan 2.5, vindenwe een langzame afname van de fout met toenemend Strouhalgetal met een macht 1

5van het Strouhalgetal. Metingen voor hogere Strouhalgetallen bleken nietvoldoendebetrouwbaar om deze trend te bevestigen voor hogere Strouhalgetallen.De afwijkingvoor twee gesuperponeerde harmonische verstoringen kan voorspeld worden door deafwijkingen van de twee afzonderlijke verstoringen bij elkaar op te tellen.

Voor extreme situaties waar pulsaties optreden, maar er geen tijdsgemiddeldestroming is, is het mogelijk dat de rotor begint te roteren als er pulsaties zijn meteen amplitude hoger dan een kritische amplitude. Deze foutieve tellingen of ”spook-tellingen” worden veroorzaakt door de vorm van de rotorbladen. Door gebruik temaken van potentiaaltheorie voor een stroming om een oneindig dunne rotorblad,kan een expliciete uitdrukking gevonden worden, die de aanvang van spooktellin-gen kan voorspellen. Door middel van experimenten is deze uitdrukking geverifieerden kan hij worden toegepast om een orde-grootte-voorspelling te doenvoor de aan-vang van foutieve tellingen. Een nauwkeurigere voorspelling kan gevonden wordendoor de stroming om een rotorblad te bestuderen. Een experimentele opstelling isgebouwd om de druk op drie verschillende locaties op een schaalmodel van de randvan een rotorblad te meten en om de stroming rondom de rand te visualiseren.Dezemetingen en visualisaties laten een redelijke overeenkomst zien met de resultaten vaneen discreet wervel-model.

Dankwoord

Het is natuurlijk een cliche te schrijven dat het proefschrift dat hier ligt niet het werkis van alleeneen persoon, maar natuurlijk is dat ook in mijn geval zeer zeker waar.En ik ben dan ook blij dat ik op deze plaats personen kan noemen die een bijdragehebben geleverd aan de totstandkoming van dit proefschrift.

In de eerste plaatst mijn promotor Mico Hirschberg.Mico, bedankt voor je begeleiding, je creativiteit en al jeideeen waarmee je me vaak hebt gemotiveerd en soms totwaanzin hebt gedreven. Met erg veel plezier heb ik al diejaren met je samengewerkt. Speciaal voor jou wilde ikdit proefschrift niet afsluiten zonder een varken.

Harry Hoeijmakers, mijn2de promotor, bedankt voorhet het grondig lezen van het concept. Dank je wel, Rinivan Dongen; voor je interesse en je raad gedurende de afgelopen jaren.

Het project waar dit proefschrift uit is voort gekomen is gefinancierddoor STW.Ik wil dan ook graag de gebruikerscommissie bedanken voor haar aandacht en input.In het bijzonder Jos Bergervoet van Elster-Instromet. Jos, bedanktvoor je medewerk-ing en voor het delen van je jarenlange ervaring met turbine debietmeters. Ook wilik graag Henk Riezebos bedanken voor zijn medewerking en Rene Peters, die mij degelegenheid heeft gegeven om twee maanden bij TNO te komen werken. Het wareneen leuke en leerzame twee maanden. Stefan Belfroid, dank je wel; twee blobbersweten meer daneen.

Ik ben erg dankbaar voor de technische steun die ik heb gekregen. Mijngebrekaan experimentele ervaring is uitstekend opgevangen door de jarenlangeervaring vanJan Willems. Bedankt Jan, het was erg prettig gebruik te kunnen maken vanal jekennis, waarmee het altijd weer mogelijk was dat in elkaar te zetten wat nodig was.Dank je wel, Freek van Uittert, voor al de tijd die je in de meetsystemen heb gestopt.Dit heeft in ieder geval als resultaat gehad dat onze opstelling de meeste computershad van alle opstellingen. Ad Holten, bedankt voor je hulp alseen van al deze com-

142 Dankwoord

puters raar deed en Freek niet aanwezig was en voor je hulp met de optica. Ook wilik Remi Zorge en Herman Koolmees bedanken voor hun medewerking. DaarnaastGerald Oerlemans; bedankt dat je bereid was alle duct tape te trotseren, toen anderetechnici niet beschikbaar waren.

Natuurlijk ben ik ook het secretariaat dank verschuldigd voor het helpen met hetadministratieve werk. Dank je wel, Brigitte. Merci, Marjan; ik heb erg veel pleziergehad met het hoteltesten en met het voorbereiden van de borrels.

Daarnaast heb ik ook het geluk gehad om samen te werken met veel studenten.Dank je wel, Wendy Versteeg, Sergio Aurajo, Jan Kuchel, Arjen Hamelinck, ErwinEngelaar, Martijn de Greef, Bram van Gessel, Floor Souren en Ineke Wijnheijmer.

De afgelopen (ruim) vier jaar waren niet zo leuk geweest zonder het gezelschapvan al mijn collega’s en oud-collega’s. Het is waarschijnlijk niemand echt ontgaan datik een liefhebber ben van koffiepauzes (of eigenlijk theepauzes), met name vanwegede discussies en gezelligheid tijdens die pauzes. Dank je wel, Marleen, Werner, Ger-ben, Geert, Ralph, Gabriel, Jieheng, Vincent, Dima, Laurens, Thijs, Ruben, Rudie,Rinie, Lorenzo, Andrzej, Alejandro, Matıas, Jurrien, Daniel, Dennis, John, David,Moasheng, Paul, Elke, Gert Jan, Willem, Herman, Gerard, Gert en de personen dieal eerder genoemd zijn.

Vrienden en familie zijn voor mij erg belangrijk geweest. Niet alleen vanwegede interesse die zij hebben getoond in mijn onderzoek, maar ook vanwege de nodigeafleiding die ze me hebben gegeven. Dank jullie wel, allemaal. In het bijzonderSaskia, Remko, Roy, Yvonne en Marijn, bedankt. Ook wil ik mijn ouders noemen:pap en mam, bedankt voor alle steun door de jaren heen.

Als laatste wil ik graag degene bedanken die me dagelijks tot steun is geweestendegene die het meest geleden heeft naast mij als het onderzoek niet goed ging. LievePatrick, dank je wel.

Curriculum Vitae

13 May 1977 Born in Heino, The Netherlands.

1989 - 1995 Stedelijk Gymnasium, Leeuwarden.

1995 - 2002 Student Mechanical Engineering,University of Twente,Engineering Fluid Dynamics Group.

• Traineeship at NASA Langley Research Center,Hampton, Virginia, USA.Application of Vortex Confinement on Unstructured Grids

• Master thesis:Aerosol Depositions in Lungsawarded the Unilever Researchprijs.

2002 - 2007 PhD Research at the Gas Dynamics Group,Department of Applied Physics,Eindhoven University of Technology.

March - May 2003 Visit at TNO Delft,Department of Fluid and Structural Dynamics.

Documents

Dynamics of turbine flow meters - Pure - Aanmeldenα angle of attack αd damping coefﬁcient m−1 β angle of rotor blade with resect to the rotor axis βav average of the angle