Dynamics of turbine flow meters
Stoltenkamp, P.W.
DOI:
10.6100/IR621983
Published: 01/01/2007
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Citation for published version (APA):
Stoltenkamp, P. W. (2007). Dynamics of turbine flow meters Eindhoven: Technische Universiteit Eindhoven DOI:
10.6100/IR621983
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Dynamics of
turbine flow meters
c
Copyright 2007
P.W. Stoltenkamp
Cover design by Oranje vormgevers
Printed by Universiteitsdrukkerij TU Eindhoven
CIP-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 924
Trefwoorden: stromingsleer / pulserende stromingen / debietmeters / meetfouten
Subject headings: flow of gases / volume flow measurements / turbine flow meters /
pulsatile flow / systematic errors
Dynamics of
turbine flow meters
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan 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 op
maandag 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. Hirschberg
en
prof.dr.ir. H.W.M. Hoeijmakers
This research was financed by the Technology Foundation STW,
grant ESF.5645
Contents
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
1.2 General description of a gas turbine flow meter . . .
1.3 Ideal rotation . . . . . . . . . . . . . . . . . . . . .
1.4 Parameter description . . . . . . . . . . . . . . . . .
1.5 Reynolds dependency of turbine flow meter readings
1.6 Thesis overview . . . . . . . . . . . . . . . . . . . .
viii
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2. Turbine flow meters in steady flow . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theoretical models of turbine flow meters . . . . . . . . .
2.2.1 Momentum approach . . . . . . . . . . . . . . . .
2.2.2 Airfoil approach . . . . . . . . . . . . . . . . . .
2.2.3 Equation of motion . . . . . . . . . . . . . . . . .
2.3 Effect of non-uniform flow . . . . . . . . . . . . . . . . .
2.3.1 Boundary layer flow . . . . . . . . . . . . . . . .
2.3.2 Velocity profile measurements . . . . . . . . . . .
2.3.3 Fully turbulent velocity profile in concentric annuli
2.3.4 Comparison of the different velocity profiles . . .
2.3.5 Effect of inflow velocity profile on the rotation . .
2.4 Wake behind the rotor blades . . . . . . . . . . . . . . . .
2.4.1 Wind tunnel experiments . . . . . . . . . . . . . .
2.4.2 Effect of wake on the rotation . . . . . . . . . . .
2.5 Friction forces . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Boundary layer on rotor blades . . . . . . . . . . .
2.5.2 Friction force on the hub . . . . . . . . . . . . . .
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46
3. Response of the turbine flow meter on pulsations with main flow . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 A basic quasi-steady model: A 2-dimensional quasi-steady
model for a rotor with infinitesimally thin blades in incompressible flow . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Practical definition of pulsation error . . . . . . . . . . . .
3.3 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Determination of the amplitude of the velocity pulsations at the location of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Acoustic model . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Synchronous detection . . . . . . . . . . . . . . . . . . . .
3.4.3 Verification of the acoustic model . . . . . . . . . . . . . .
3.4.4 Measurements of velocity pulsation in the field . . . . . . .
3.5 Determination of the measurement error of the turbine meter . . . .
3.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Dependence on Strouhal number . . . . . . . . . . . . . . .
3.6.2 Dependence on Reynolds number . . . . . . . . . . . . . .
3.6.3 High relative acoustic amplitudes . . . . . . . . . . . . . .
3.6.4 Influence of the shape of the rotor blades . . . . . . . . . .
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
49
50
4. Ghost counts caused by pulsations without main flow . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Onset of ghost counts . . . . . . . . . . . . . . . . . . . . .
4.2.1 Theoretical modelling of ghost counts . . . . . . . .
4.2.2 Experimental setup for ghost counts . . . . . . . . .
4.2.3 Experiments . . . . . . . . . . . . . . . . . . . . .
4.2.4 Comparing measurements with results of the theory .
4.3 Influence of vibrations and rotor asymmetry . . . . . . . . .
4.3.1 Vibration and friction . . . . . . . . . . . . . . . . .
4.3.2 Rotor blades with chamfered leading edge . . . . . .
79
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87
89
91
93
93
93
2.6
2.7
2.5.3 Tip clearance . . . . . . . . . . . . . . . . . . . . . .
2.5.4 Mechanical friction . . . . . . . . . . . . . . . . . . .
Prediction of the Reynolds number dependence in steady flow
2.6.1 Turbine meter 1 . . . . . . . . . . . . . . . . . . . . .
2.6.2 Turbine meter 2 . . . . . . . . . . . . . . . . . . . . .
2.6.3 Effect of tip clearance . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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vii
4.4
4.5
Flow around the edge of a blade . . . . . . . . . . . . . . . . . . .
4.4.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . .
4.4.2 Experimental set up for flow around an edge . . . . . . . .
4.4.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Comparing measurements with results of the numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Conclusions . . . . . . . . .
5.1 Introduction . . . . . . . . . .
5.2 Stationary flow . . . . . . . .
5.3 Main flow with pulsations . . .
5.4 Pulsations without main flow .
5.5 Recommendations . . . . . . .
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Appendix
94
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95
98
102
104
109
109
109
110
111
111
113
A. Mach number effect in temperature measurements . . . . . . . . 115
B. Boundary layer theory . . . . . . . . . . . . . . . .
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Blasius exact solution for boundary layer on a flat plate . .
B.3 The Von Kármán integral momentum equation . . . . . . .
B.4 Description laminar boundary layer . . . . . . . . . . . .
B.5 Description turbulent boundary layer . . . . . . . . . . . .
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C. Measurements . . . . . . . .
C.1 Introduction . . . . . . . . . .
C.2 Pulsation frequency of 24 Hz
C.3 Pulsation frequency of 69 Hz
C.4 Pulsation frequency of 117 Hz
C.5 Pulsation frequency of 363 Hz
C.6 Pulsation frequency of 730 Hz
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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
c0
speed of sound
f
frequency
hblade height of a rotor blade
equation 3.25
m s−1
Hz
m
m−1
k
wave number
m′
mass flow
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
tblade blade thickness
kg s−1
m or s
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
m2
A
cross-sectional area
B′
total specific enthalpy
D
pipe diameter
E
relative deviation from ideal rotation
equation 2.14
Epuls relative error caused by periodic pulsations
equation 3.11
m2 s−2
m
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
K
kg m2
m3 rad−1
meter factor
Lblade chord length of a rotor blade
m
Lhub
length of the hub in front of the rotor
m
Q
volume flow
R
root-mean-square radius
S
pitch or area
m3 s−1
q
2 +r 2
rin
out
2
m
m or m2
xi
T
temperature or period of the pulsations
K or s−1
Tmech mechanical friction torque
kg m2 s−2
Tair
air friction torque
kg m2 s−2
T bf
torque imposed on the fluid by the body
kg m2 s−2
Td
driving torque
kg m2 s−2
Tf
total friction torque
kg m2 s−2
U
mean velocity in the annulus in front of the rotor
V
volume
W
width of the rotor
m s−1
m3
m
Greek symbols
◦
α
angle of attack
αd
damping coefficient
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
m2 s−1
φm
mass flow
kg s−1
Γ
circulation
m2 s−1
γ
Poisson’s ratio
µ
dynamic viscosity
ν
kinematic viscosity
m2 s−1
ω
rotation speed
rad s−1
ωid
ideal rotation speed
rad s−1
m−1
kg m−1 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−1 s−2
τw
shear stress at the wall
kg m−1 s−2
Dimensionless numbers
CD
drag coefficient, FD /( 12 ρu2 A)
′
CD
drag coefficient, FD /( 12 ρu2 wt)
CL
lift coefficient, FL /( 21 ρu2 A)
He
Helmholtz number,
M
Mach number,
Pr
Prandtl number, ν/a with a the thermal diffusivity
Re
Reynolds number,
Sr
Strouhal number,
fL
c0
u
c0
uL
ν
fL
u
1
Introduction
1.1
Introduction
In industry axial turbine flow meters are used to measure volume flows of gases and
liquids. They are considered reliable flow meters and at suitable conditions can attain
high accuracies in the order of 0.1% for liquids and 0.25% for gases. An accuracy
up 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 applications, for example in the chemical, petrochemical, food and aerospace industry. The
internal diameter of these flow meters can vary from very small, e.g. 6 mm, to very
large, e.g. 760 mm.
In the Netherlands gas turbine flow meters are commonly used to measure natural
gas flow. Because the Netherlands transported in 2005 95.2 billion m3 of natural gas,
small systematic measurement errors can lead to over- or underestimation of large
volumes of natural gas. This makes the accuracy of flow meters crucial at all flow
conditions. A new development is the exploration of the possibility to correct flow
measurements for non-ideal flow conditions on the basis of a physical model for the
response 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 this
drawing the most important elements of a turbine flow meter are given. Turbine flow
meters are placed in line with the flow. Sometimes they are placed in measuring manifolds, where several flow meters are placed in parallel streams, in order to increase
the overall dynamic range of the set up. Usually the flow passes first through a flow
2
1. Introduction
C
A
B
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 uniform
flow. Subsequently, the flow is forced through an annular channel and through the rotor (B), see also figure 1.2. The blades of the rotor are often flat plates or have a helical
shape. 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 the
rotor. The most common detection methods are mechanical detection and magnetic
detection. Mechanical detection of the rotor speed is measured by transferring the
rotor 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 a
designated 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 ElsterInstromet. The dynamical response measurements have been carried out at the Eindhoven University of Technology with the gas turbine meter type SM-RI-X G250, see
figure 1.3. This meter has an internal pipe diameter of 100 mm. The accuracy of the
Figure 1.3: Photograph of the SM-RI-X G250 turbine flow meter (by courtesy of ElsterInstromet).
flow measurement is 0.1% for volume flows in the range from 20 to 400 m3 /h. The
meter is designed for pressures ranging from atmospheric pressure up to 20 bar (this
type of meter is also available for work pressures up to 100 bar). The rotor is made
of aluminium and has helical shaped blades (see figure 1.2). We will refer to this meter as turbine meter 1. Additional steady flow experiments have been performed by
Elster-Instromet with simplified prototypes which we refer to as turbine meter 2, 3, 4
and 5. Additional experiments with oscillatory flow have been performed by Gasunie
with a larger version of the SM-RI-X G250, the SM-RI-X G2500 with a internal pipe
diameter of 300 mm.
4
1. Introduction
u
u
in
re l
u
b
o u t,x
w r
= u
in
b
w r
Figure 1.4: Steady flow entering and leaving the rotor for an ideal frictionless rotor with
infinitesimally thin helical rotor blades with blade angle β.
1.3
Ideal rotation
When ideal rotation is considered, it is assumed that the flow through the turbine
meter is uniform, incompressible and steady, that the rotor rotates with no friction
and that the rotor is shaped as a perfect helix with infinitesimally thin blades. Under
these circumstances the rotation speed of the rotor is determined by the pitch of the
rotor, S, defined by:
S=
2πr
,
tan β
(1.1)
with r the radius of the rotor and β the angle of the rotor blades with respect to
the rotor axis (see figure 1.4). In an ideal case the pitch corresponds to the axial
displacement of the fluid during one revolution of the rotor. For a perfect helicoidal
rotor 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 rotor
is parallel to the blades of the rotor. This means that the inlet velocity and the rotation
velocity are related through the angle of the rotor blades, β, as:
ωid r
= tan β ,
uin
(1.2)
1.4. Parameter description
5
with ωid the angular velocity of the rotor for the ideal situation considered and uin is
the velocity of the flow entering the rotor. The angular velocity in this ideal situation
is
ωid =
2πuin
uin tan β
=
.
r
S
(1.3)
Because the volume flow, Q, is equal to the inflow velocity multiplied by the crosssectional area of the rotor, i.e. Q = uin A, we find a relationship between the volume
flow and the rotational speed:
Q=
AS
ωid .
2π
(1.4)
This relationship is applied in an actual turbine flow meter in the form:
Q = Kωid ,
(1.5)
where K is called the meter factor, which is determined by calibration. Ideally, K
should be a constant.
1.4
Parameter description
In principle for steady flow the meter factor K of a specific meter depends on dimensionless parameters such as:
• the Reynolds number Re =
• the Mach number M =
uin L
ν
uin
c0
• the ratio of mechanical friction torque, Tmech , to the driving fluid torque
Tmech
R3 ρu2in
where L is a characteristic length such as the blade chord length, ν is the kinematic
viscosity of the fluid, c0 is the speed of sound R is the root mean square radius of
the rotor and ρ the fluid density. The manufacturer uses steady flow calibrations at
different pressures to distinguish between Reynolds number effects and the influence
of mechanical friction. In general the Mach number dependency is a small correction
due to a Mach number effect in the temperature measurements at high flow rates (see
Appendix A).
In this thesis we will consider unsteady flow. In such case the response of the
meter will also depend on:
• the Strouhal number Sr =
fL
uin
6
1. Introduction
• the amplitude of the perturbations
|u′in |
uin
• The ratio of fluid density, ρ, and rotor material density, ρm , i.e.
ρ
ρm
where f is the characteristic frequency of flow perturbations and |u′in | is the amplitude 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. In
reality friction forces and drag forces cause the rotor to rotate at a rotation speed that
differs from the rotational speed of the ideal rotor. The difference between the actual
rotor speed and the ideal rotor speed is known as rotor slip. Because the drag forces
depend on flow velocity and the viscosity of the medium, the rotor slip depends on
Reynolds number, Re. A meter designer tries to make the volume flow measured by
the meter to be a function that is as linear as possible in terms of the rotational speed
for a dynamic range of at least 10:1. With every meter the manufacturer provides a
calibration, that gives the rotor slip as function of the Reynolds number or sometimes
as function of the volume flow. This calibration is unique for every meter due to the
sensitivity of the meter to small manufacturing differences or differences caused by
damage or wear. One of the aims of the designer is to reduce this sensitivity of the
meter factor, i.e. the quantity K, for manufacturing inaccuracies, damage or wear.
1.6
Thesis overview
In this thesis, the behaviour of turbine flow meters is investigated experimentally aiming at development of physical models allowing corrections for deviations from ideal
flow.
In chapter 2 the Reynolds number dependence of the turbine flow meter is investigated analytically. The driving torque on the rotor is obtained by using conservation
of momentum on a two-dimensional cascade of rotor blades. Using the equation of
motion of the rotor, its rotation speed is determined. We use in this chapter a theoretical model developed by Bergervoet (2005) which we extent by considering the
influences of non-uniform flow and drag forces. The effect of the inlet velocity profile
is investigated using models and measurements. The effect of several friction forces
is modelled analytically. The last part of this chapter compares the model with calibration measurements obtained by Elster-Instromet for several turbine flow meters.
Chapter 3 studies the effect of pulsations superimposed on main flow. Pulsation
can 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 investigate the applicability of this model. A detailed description is given of the measurement set up and measurements methods. Finally, the results are discussed.
Chapter 4 deals with the extreem case of chapter 3, where the flow is purely
oscillatory and there is no main flow. This can induce the rotor to rotate and measure
a flow while there is no net flow. We call this ghost counts or spurious counts. The
first part of this chapter describes two physical models to predict the onset of ghost
counts. The models are compared with experiments. The second part of this chapter
investigates the flow around the edge of a rotor blade in pulsating flow. First, this
investigation is carried out experimentally. These results are compared with a discrete
vortex 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 meter
in steady flow. The development of a theoretical model describing the behaviour of a
turbine flow meter has been endeavoured experimentally and analytically for a long
time (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 understand the behaviour of turbine flow meters use numerical methods to compute the
flow field in a turbine flow meter (von Lavante et al. (2003), Merzkirch (2005)). A
theoretical model allows the investigation of, for example, meter geometry, making it
possible to develop better design criteria, or to assess the influence of different fluid
properties. Rather than considering a numerical method we will consider an extension of the more global analytical model as proposed by Thompson and Grey (1970).
Our global model aims at understanding important phenomena in the behaviour of
turbine flow meters. Since in practice deviations in the dependence on Reynolds
number of 0.2% are significant, we do not expect to succeed in making such accurate
predictions of the deviations. We try to obtain some insight into the problem of the
design of a flowmeter.
The turbine meter is modelled using the equation of motion for the rotor. The
flow passing through the rotor induces a driving torque, Td , on the rotor. First, two
approaches to obtain this driving torque will be discussed. Next, the influence of
the inlet velocity, uin at the front plane of the rotor will be investigated by using a
boundary layer description, actual velocity measurements in a dummy of a turbine
flow meter and a model for fully developed flow. Wind tunnel measurements have
been 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 and
their individual effect on the rotation speed of the rotor will be shown. In the last
part of this chapter the model is applied to different turbine flow meters at different
Reynolds numbers and the results are compared to calibration measurements provided 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 calculate
the driving torque on the rotor. One of the main limitations of this method is that full
fluid guidance is assumed. It is assumed that there is a uniform flow tangential to
the rotor blades at the rotor outlet. This assumption is only true for rotors with high
solidity. This implies a gap between successive blades, which is narrow compared to
the blade chord length. Weinig (1964) showed, using potential flow theory for a twodimensional planar cascade, that the ratio of the gap between the blades and blade
length (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 rotor
by using airfoil theory to obtain the lift coefficient of an isolated rotor blade. With
this approach there is no assumption of full fluid guidance, but blade interference
is ignored. This means that increasing the number of blades would always increase
the lift force proportionally. Thompson and Grey (1970) improved this approach by
using the two-dimensional planar cascade theory of Weinig (1964) to account for the
interference effects.
Both the integral momentum method and the airfoil method will be explained in
more detail in the following sections. We later actually use only the integral momentum method, which has been used earlier in simplified form by Bergervoet (2005) at
Elster-Instromet.
2.2.1
Momentum approach
The turbine meter is a complex three-dimensional flow device (see figure 2.1). As an
approximation this three-dimensional problem will be treated as a two-dimensional
infinite cascade of rotor blades with uniform axial flow, uin , at radius r as approximation of the flow inside an annulus between r and r + dr. The x-direction refers to the
axial direction. The y-direction refers to the azimuthal direction (see figure 2.2). The
radial velocity is neglected and constant rotation with a rotational angular velocity ω
is assumed. To obtain the torque on the rotor we will integrate over the blade length
in radial direction. The control volume enclosing the rotor is shown in figure 2.2.
2.2. Theoretical models of turbine flow meters
h
r
r
tip
11
b la d e
h u b
x
q
r
w
d r
Figure 2.1: The rotor of the turbine flow meter. We assume that the flow in an annulus between r and r + dr behaves as the flow in a two-dimensional infinitely long
cascade shown in figure 2.2.
To calculate the driving torque on the rotor, the integral mass conservation law and
integral momentum equation is used for this two-dimensional cascade of blades:
ZZ
ZZZ
d
(2.1)
ρv · ndA = 0 ,
ρdV +
dt CV
CS
ZZ
ZZ
ZZ
ZZZ
d
ρv (v · n) dA = −
pndA +
τ ndA + F bf ,(2.2)
ρvdV +
dt CV
CS
CS
CS
applied to a fixed control surface CS enclosing the rotor, this surface has an outer
normal n, the fixed control volume within CS is denoted as CV , ρ is the fluid density,
v is the velocity vector, p is the pressure, τ is the viscous stress tensor and F bf are
the forces imposed on the fluid by the turbine.
Full fluid guidance is assumed; the flow leaves the rotor with a velocity parallel to
the blades along the whole circumference (or the y-direction in our 2D model, figure
2.2). This implies that we neglect radial velocities and the effect of the Coriolis
forces. We assume that the flow enters the rotor without any azimuthal velocity, vθ =
0 (in a two dimensional representation vy = 0). Assuming steady incompressible
flow and applying the conservation of mass (equation 2.1) to a volume element of
height dr (figure 2.1), we get:
uin,x dAin = uout,x dAout ,
(2.3)
12
2. Turbine flow meters in steady flow
w r
W
b
u
u
u
in
re l
b
w r
u
o u t,x =
o u t,y
in
n
re l
t
n
u
u
L
C S
b la d e
y
x
Figure 2.2: Flow entering and leaving the cascade representing the rotor in an annulus between r and r + dr.
where uin,x and uout,x are x-component of the the incoming and outgoing velocity,
respectively, and dAin and dAout are the inflow area and the outflow area, respectively. If the inflow and outflow area are assumed to be equal and the flow is incompressible, dAin = dAuit = 2πrdr, so that the x-component of the incoming velocity
is 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 viscous
forces, Re >> 1, the momentum equation in the y-direction for a steady flow through
an element dr becomes:
ρ ((uout,y + ωr) uout,x dAout − uin,x ωrdAin ) = dFbf,y ,
(2.4)
From the velocity diagram in figure 2.2 it can be seen that:
uout,y = uout,x tan β − ωr .
(2.5)
2.2. Theoretical models of turbine flow meters
13
Substituting equations 2.3 and 2.5 in equation 2.4, the y-component of the force
imposed by the rotor on the fluid, dFbf,y is found:
dFbf,y = ρu2out,x tan βdAout − uin,x ωrdAin .
(2.6)
The force of the fluid on the rotor is opposite and equal to the force of the rotor on the
fluid, dFbf,y = −dFf b,y . The torque exerted by the fluid element on the rotor axis,
dTd , is estimated to be:
dTd = rdFf b,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:
Z rtip
Z rtip
ρuin,x ωr2 dAin .
(2.8)
ρu2out,x (tan β)rdAout +
Td = −
rhub
rhub
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 radius r and thickness dr is approximated as an
infinite two-dimensional cascade of rotor blades (see figure 2.3). In contrast to the
momentum approach there is no assumption that flow is attached. The driving torque
on the rotor blade is now evaluated by determining the lift and drag forces on the rotor
blades in a coordinate system fixed to the blade. The lift force, FL , acts perpendicular
to the relative inlet velocity, uin,rel = (uin,x , ωr), and the drag force, FD acts parallel
to this inlet velocity. The y-component of the force of the flow on the blade can now
be expressed in terms of lift, FL , and drag, FD ;
Fy = n (−FL cos φ + FD sin φ) ,
(2.9)
where φ = β − α = arctan uωr
, with β the angle of the rotor blade (with respect
in,x
to the x-axis), n is the number of blades and α the angle of attack of the incoming
flow. The lift- and drag coefficient are defined as:
CL =
FL
1
2
2 ρuin,rel Lblade
,
FD
CD = 1 2
,
2 ρuin,rel Lblade
(2.10)
where Lblade is the chord of the blade. The lift and drag coefficients are functions
of the angle of attack, α, depend weakly on Reynolds number and on Mach number.
14
2. Turbine flow meters in steady flow
F
a
u
in ,r e l
f
u
w r
in
F
F
D ,y
f
L ,y
D
F
L
b
t
L
b la d e
y
x
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 with n blades can be written as:
Z rtip
1
Td =
nρu2in,rel Lblade (−CL cos φ + CD sin φ) rdr .
(2.11)
rhub 2
2.2.3
Equation of motion
The driving torque, Td , is known from equation 2.8 or 2.11. To determine the angular
velocity, ω, of the rotor, the equation of motion of the rotor is used:
dω
= Td − Tf ,
(2.12)
dt
where Irotor is the moment of inertia of the rotor and Tf is the friction torque on
the rotor, assuming a quasi-steady flow through the rotor. Using equation 2.8 or 2.11
Irotor
2.2. Theoretical models of turbine flow meters
15
for the torque implies that we assume a quasi-steady flow through the rotor. In this
chapter we investigate the rotor in steady rotation, for which the equation of motion
reduces to:
Td = T f .
(2.13)
The different friction forces will be discussed in the following sections. This equation
can be used to predict the steady rotation speed of the rotor, ω. By comparing this
rotation speed with the ideal rotation speed, ωid (see equation 1.3), the deviation of
the 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 turbine meter can be estimated.
In the following sections the analysis will be applied using the momentum approach (equation 2.8) to two types of turbine flow meters. The first one, referred to
as turbine meter 1, is the Instromet SM-RI-X G250 with a diameter of D = 0.1 m
used in the experiments at the set up in Eindhoven. The second one is a simplified
turbine meter with diameter of D = 0.2 m, this rotor will be referred to as turbine
meter 2. The second turbine meter has a simplified geometry. An example of this
simplification is the geometry at the rotor tip (see section 2.5.3). This simplified geometry should allow a better comparison of experiment with the theory. Information
about the geometry of the two flow meters is given in table 2.1 The chord length of
the rotor blades of turbine meter 1 can be calculated using:
W
,
(2.15)
cos β(r)
the angle of the blade relative to the rotor axis. The blades
with β = arctan 2πr
S
of the second turbine meter, turbine meter 2, are reduced at the tip to a chord length
of Lblade (rtip ) = 0.035 m. The chord length of the rotor blades of this turbine meter
can be written as:
Lblade (r) =
Lblade (r) = Lblade (rhub ) +
Lblade (rtip ) − Lblade (rhub )
(r − rhub ) .
hblade
(2.16)
In the following sections the effect of non-uniform flow, the blade drag and other
friction forces are investigated separately, the deviation from the ideal rotation is
calculated for several flows up to Qmax as indicated for the meter. Two scenarios
were followed; in the first scenario the calculations were done using the properties
of air at 1 bar (absolute pressure), ρ = 1.2 kg/m3 and ν = 1.5 × 10−5 m2 /s, and
16
2. Turbine flow meters in steady flow
pipe diameter, D (m)
blade thickness, t (mm)
number of blades, n
rhub /D
rout /D
S/D
W/D
hblade /D = (rtip − rhub )/D
Lhub /D
turbine meter 1
0.1034
1.6
16
0.360
0.500
2.704
0.213
0.140
0.763
turbine meter 2
0.2030
4
14
0.250
0.500
3.941
0.148
0.240
1.049
Table 2.1: Dimensions of the two turbine flow meters used in the calculations, where rhub 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 (equation 1.1), W is the width of the rotor, hblade
is the height of the blade (span of the blades) and Lhub is the length of the hub in
front of the rotor. Except for the blade thickness t and the number of blades n, all
values are made dimensionless with the diameter, D.
in the second scenario the properties of natural gas at 9 bar (absolute pressure) were
used, ρ = 7.2 kg/m3 and ν = 1.5 × 10−6 m2 /s. These conditions correspond to
the test conditions used by Elster-Instromet. The resulting deviation, E, is plotted
against the Reynolds number, Re = U Lblade /ν, where Lblade is the length of a rotor
blade measured at the tip and U 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 approximation, if the ratio of the distance between the blades and blade length is sufficiently
small, 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. However, the departure from full fluid guidance is expected to be small. Using the theory
of (Weinig, 1964), we estimate that the tangential velocity uout,y will be about 5%
smaller than the tangential velocity for full fluid guidance. The reduction in the tangential velocity decreases the driving torque exerted by the flow on the rotor and this
decreases 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 entering
the 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 due to
upstream bends. Parchen (1993) and Steenbergen (1995) showed that swirl decays
extremely slowly. Swirl can have effect the accuracy of turbine meters (Merzkirch,
2005). Properly designed flow straighteners as designed by Elster-Instroment placed
in front of a turbine flow meter reduce the effect of swirl considerably. Therefore in
the calculation we assume that there is no azimuthal velocity (no swirl). We limit our
discussion to the non-uniformity of the axial velocity, uin (r).
Thompson and Grey (1970) predicted that the inlet velocity profile plays an important role in the rotation speed of the rotor.
The influence of the velocity profile entering the rotor will be investigated in this
section. The shape of the velocity profile entering the rotor is first calculated using
boundary layer theory. Velocity profile measurements carried out in a dummy of a
turbine meter will be compared with the boundary layer theory and a fully developed
turbulent annulus flow assumed by Thompson and Grey (1970). The rotation rate of
a rotor for velocity profile based on boundary layer theory and for a measured flow
profile 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 continues through
an annular pipe segment of length Lhub 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. At
the leading edge of the hub a new boundary layer starts to form on the hub and on the
outer wall. The velocity profile is assumed axisymmetric and can be divided in three
regions (see figure 2.4). The first region is the boundary layer on the hub. The second
region is the region between the boundary layers, where the velocity is approximately
uniform. 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 optimal
conditions around a Reynolds number of ReLhub ≈ 3 × 105 (Schlichting, 1979). This
would imply that there is a significant laminar part of the boundary layer on the hub
even for ReLhub > 3 × 105 . However, we will assume that above a critical Reynolds
number the boundary layer is turbulent from the start, ignoring the effect of transition.
The boundary layer thickness is calculated using the von Kármán integral momentum equation (see Schlichting (1979)). Appendix B provides a brief discussion
of boundary layer theory. The von Kármán equation obtained by integration of the
mass and momentum equations over the boundary layer is:
dU
τw
d
U 2 δ2 + δ1 U
=
,
dx
dx
ρ
(2.17)
18
2. Turbine flow meters in steady flow
o u te r w a ll
r
II
u (r)
I
d (x )
ro to r
s tra ig h te n e r
III
h u b
h
b la d e
r
h u b
L
o u t
W
h u b
r
x
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 (for
definition see equation B.3), δ2 the momentum thickness (for definition see equation
B.4) and τw the shear stress at the wall. For the calculation of the laminar boundary
layer, a third order polynomial description of the boundary layer profile is used in
combination with Newton’s law for τw (see Appendix B). This was found to be an
accurate description of a laminar boundary layer by Pelorson et al. (1994) and Hofmans (1998). For turbulent flow the boundary layer is described using a 1/7th power
law description for the velocity profile combined with the empirical law of Blasius
for the wall shear stress (see Appendix B). Using these models, the displacement
thickness, δ1 , the momentum thickness, δ2 , and the shear stress at the wall, τw , are
calculated 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. Using
the 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)
where Q is the volume flow, rout is the radius of the outer wall and rhub is the radius
of the hub. The boundary layers on the outer pipe wall and on the hub are assumed
to have the same thickness.
The velocity profile in front of the rotor of a turbine meter with geometrical dimensions equal to the turbine meter 1, is calculated. This meter has a radius of the
outer wall, rout = 0.050 m and a radius of the hub rhub = 0.037 m. The hub length
in front of the rotor is Lhub = 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 turbulent boundary layers the velocity profile is plotted in figure 2.5(b). As expected the
2.3. Effect of non-uniform flow
19
1
0.8
rhub
u/umax
0.6
0.4
ReL = 3 X103
4
ReL hub = 1.2 X 10
4
ReL hub = 3.9 X 10
Re hub = 1.4 X 105
0.2
L
hub
0
0.7
0.75
0.8
0.85
r/r
0.9
0.95
1
out
(a) laminar boundary layers
1
0.8
rhub
u/umax
0.6
0.4
ReL = 3 X103
ReL hub = 1.2 X 104
ReLhub = 3.9 X 104
Re hub = 1.4 X 105
0.2
L
hub
0
0.7
0.75
0.8
0.85
r/rout
0.9
0.95
1
(b) turbulent boundary layers
Figure 2.5: Velocity profile entering the rotor for turbine meter 1 with a diameter 0.1034 m
calculated using boundary layer theory. The velocity, u, divided by the maximum velocity, umax is plotted against the radius for different Reynolds numbers
(Re = U Lhub /ν = 3 × 103 , 1.2 × 104 , 3.9 × 104 and 1.4 × 105 . (a) shows the velocity profile with laminar boundary layers, (b) the velocity profile with turbulent
boundary layers.
20
2. Turbine flow meters in steady flow
laminar boundary layers are thinner than the turbulent boundary layers. The velocity
profile for turbulent boundary layers is more uniform than that for laminar flow.
2.3.2
Velocity profile measurements
To examine whether the boundary layer description of the velocity profile is an adequate approximation of the velocity profile, measurement were carried out with a
hot wire anemometer and a Pitot tube in the set up described in section 3.3. In this
set up turbine flow meter 1 with a diameter of D = 0.1 m, is placed at the end of
a pipe with a length of more than 30 times its diameter. The pipe flow is supplied
by a high pressure dry air reservoir (60 bar). A choked valve is controlling the mass
flow through the pipe. In order to measure the velocity profile just upstream of the
rotor, the flow meter was replaced by a dummy. The dummy is a replica of the forward part of the meter, including the flow straightener, up to the rotor. The remainder
of the flow meter, including the rotor, has been removed providing easy access for
the measurement probes. The Pitot tube has a diameter of 1 mm and is connected
to 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 10 s measurement at
a sample frequency of fs = 10 kHz. Before measuring the velocity profile just in
front the rotor (but in absence of the rotor), the velocity profile in the pipe upstream
of the turbine flow meter was measured using the Pitot tube. Measurements were
performed at four different velocities in the pipe, 2, 4, 10, 15 m/s. The measured
profiles are plotted in figure 2.6. The Reynolds number, ReD , mentioned in figure 2.6
is 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 developed
turbulent pipe flow.
Measurements of the annular flow 1 mm downstream of the dummy of the forward part of the meter were performed at seven different average velocities in the pipe
(0.5, 1, 1.5, 2, 4, 10 and 15 m/s), resulting in Reynolds numbers, Re = U Lhub /ν,
where Lhub is the length of the hub in front of the rotor (see figure 2.4) and U
the mean velocity in the annulus outside the boundary layers (equation 2.18). This
Reynolds number ranges from 3.0 × 103 up to 1.5 × 105 . From the measurements
shown in figure 2.7, it can be seen that the velocity profile is asymmetric. The
asymmetry is increasing with increasing Reynolds number. It has a maximum velocity closer to the outer wall than to the hub. For lower Reynolds numbers near
the walls the velocity profile resembles the laminar boundary layer velocity profile,
for Reynolds number above 104 the velocity profile resembles more the turbulent
2.3. Effect of non-uniform flow
21
1
0.9
0.8
0.7
u/umax
0.6
0.5
0.4
ReD = 1.5 X 104
0.3
4
ReD = 2.8 X 10
0.2
4
ReD = 7.1 X 10
0.1
0
−0.5 −0.4 −0.3 −0.2 −0.1
5
ReD = 1.1 X 10
0
r/D
0.1
0.2
0.3
0.4
0.5
Figure 2.6: The velocity profile in the pipe just upstream of the turbine flow meter, measured
at four different Reynolds numbers, ReD = umax D/ν = 1.5 × 104 , 2.8 × 104 ,
7.1 × 104 and 1.1 × 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 1 mm downstream
of the dummy of the turbine meter. At this point there is a flow for r/rout > 1,
because of entrainment of air in the airjet flowing out of the dummy (figure 2.9). We
therefore observe some velocity at the location of the pipe wall, r/rout = 1, where
in 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 has been
studied in literature, because of the many engineering applications and in order to
obtain fundamental insight in turbulence. Brighton and Jones (1964) found experimentally that the position of the maximum velocity of such fully developed flows is
closer to the inner wall than to the outer pipe wall. The position depends on Reynolds
number and ratio rhub /rout of the inner wall radius, rhub , and the outer wall radius,
rout . The results found here differs in that respect.
22
2. Turbine flow meters in steady flow
1
0.8
rout
r
hub
u/umax
0.6
0.4
Re = 3.1 X 103
0.2
Re = 5.6 X 103
4
Re = 1.2 X 10
0
0.65
0.7
0.75
0.8
0.85
r/rout
0.9
0.95
1
1.05
(a) Re < 2 × 104
1
0.8
rout
rhub
u/umax
0.6
0.4
Re = 2.0 X 104
Re = 3.9 X 104
0.2
4
Re = 9.2 X 10
5
Re = 1.4 X 10
0
0.65
0.7
0.75
0.8
0.85
r/rout
0.9
0.95
1
1.05
(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 1 mm downstream of a dummy of the
forward part of the meter. The velocity, u, normalised by the maximum velocity,
umax as a function of the radius for four different Reynolds numbers.
2.3. Effect of non-uniform flow
23
p ro b e
o u te r w a ll
r
o u t
s tra ig h te n e r
u (r)
h u b
L
1 m m
h u b
r
h u b
r
x
Figure 2.8: Schematic drawing of the position of the hot wire during the velocity measurements.
p ip e w a ll
flo w e n tra in m e n t
m a in (je t) flo 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 measured profile (figure 2.7) and the profile of a fully developed turbulent flow as found
by Brighton and Jones (1964) are quite different. Comparing the result of the boundary layer calculations for turbine flow meter 1 with the measurements in the same
meter, the measured profiles show a clear asymmetry dependent on the Reynolds
number. Fully developed turbulent flow in an annular channel (e.g. Brighton and
Jones (1964)) displays a maximum velocity closer to the inner wall than to the outer
wall. However, in our measurements the maximum velocity is closer to the outer
wall. This indicates that the measured velocity profile does not resemble the fully
developed turbulent flow in an annulus. This is not surprising, since the length of
the hub, Lhub , is relatively short, Lhub ≈ 5.5(rout − rhub ). The asymmetry in the
24
2. Turbine flow meters in steady flow
measured profile can be caused by flow separation at the front of the hub, resulting
in a velocity profile with higher velocity along the outer wall (see figure 2.10). The
observed velocity maximum would be due to the flow separation at the sharp edge of
the nose of the hub. Similar behaviour is observed downstream of a sharp bend in a
pipe.
o u te r w a ll
r
o u t
r
h u b
ro to r
s tra ig h te n e r
u
h u b
L
h u b
r
x
Figure 2.10: Flow is expected to separate at the leading edge of the hub causing the flow to
accelerate 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 , the
driving torque is calculated using the predicted velocity profile based on boundary
layer theory. The mechanical friction forces, the fluid friction and the thickness of the
blades are ignored. The results are compared to the calculation of the driving torque
predicted for a uniform velocity. As we assume incompressible flow, the continuity
equation gives that the incoming velocity is equal to that of the axial component of
the outgoing velocity, uin = uin,x = uout,x . The momentum equation (equation 2.8)
reduces to:
Z rtip
Td = −
ρuin (uin tan β + ωr) 2πr2 dr .
(2.19)
rhub
For steady flow and in absence of friction the equation of motion of the rotor (equation 2.13) reduces to: Td = 0. For a given geometry of the rotor and a known
incoming velocity profile, the rotation speed of the rotor can then be calculated. As
the velocity profile depends on the Reynolds number, Re = U Lhub /ν, the deviation
of the rotation speed from ideal rotation speed for a uniform inflow, E (see equation
2.14), is plotted against Reynolds number. In figure 2.11 the deviation in rotation
speed has been plotted for the laminar and turbulent boundary layer profiles (figure
2.5) and for the measured profile (figure 2.7). Compared to a uniform flow the rota-
2.4. Wake behind the rotor blades
25
8
velocity profile measurements
turbulent boundary layers
laminar boundary layers
E = (ω − ωid) / ωid * 100 (%)
7
6
5
4
3
2
1
0 3
10
4
5
10
10
6
10
Re
L
hub
Figure 2.11: The deviation of the rotation speed, ω, from the rotation speed for a uniform
inlet velocity profile, ωid versus Reynolds number, Re = U Lhub /ν. Turbulent
boundary layer approximation (solid line), laminar boundary layer approximation (dashed line) and the measured velocity profile (◦).
tion speed of the rotor increases in the order of one percent for a velocity profile based
on laminar or turbulent boundary layer theory. The turbulent boundary layer causes
the rotor to rotate faster than the laminar boundary layers. The measured velocity
profile induces much larger deviations. As we are aiming for an accuracy of 0.2%, it
is clear that the velocity profile plays a very significant role in the rotation speed of
the rotor, as already observed by Thompson and Grey (1970). In further calculations
discussed in this chapter, the boundary layer model is used. We have to keep in mind
that 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 flow
also exerts a drag force on the rotor. The effect of the forces caused by the pressure
difference between the pressure and the suction side of the rotor blade and by the
friction of the fluid on the solid surface of the blades (described in section 2.5.1) can
be included in the momentum conservation balance described in section 2.2.1. To
26
2. Turbine flow meters in steady flow
include the pressure drag, a model for the wake is proposed. In this model we will
assume that the wake of the blade in the rotor has the same structure as for a single
isolated blade in free stream (see figure 2.12).
u
n
in ,r e l
u
ro to r b la d e
A
f
w a k e
m
C S
o u t,r e l
w
w a k e
A
Figure 2.12: Wake behind a rotor blade.
Betz, Prandtl and Tietjens (1934) found that it is possible to calculate the drag
force on a body in an unbounded uniform flow by applying a momentum balance on
a large control surface surrounding the body. The control volume is chosen around
the rotor blade, with a control surface CS with a normal vector n as shown in figure
2.12. The control volume has to be chosen far from the body. There, the streamlines
in the flow are again approximately parallel and the pressure over the wake can be
considered uniform and equal to the pressure of the uniform flow. The rotor blade and
the wake cause a displacement of the flow over the sides. We apply the momentum
equation on this controle volume for steady incompressible flow.
Assuming that outside the wake the velocity, u, can be approximated by the free
stream velocity, u∞ = uin , this equation reduces to:
Z
uout (u∞ − uout ) dy = ρu2∞ δ2,wake ,
(2.20)
FD = ρ
wake
where the integral can be limited to the wake, because uout = u∞ outside the wake
and δ2,wake is the momentum thickness of the wake. With this equation the drag
FD
′ = C Lblade =
,
coefficient of a blade of length Lblade and thickness t, CD
D
1
t
ρu2
t
2
in,rel
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 the wake
′ =0
equal to the pressure of the uniform main flow, the drag coefficient vanishes CD
(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 pressure
and a drag is experienced by the blade. The flow just behind the blade is extremely
2.4. Wake behind the rotor blades
27
complex. We will therefore consider the wake at some distance from the trailing edge
of the blade.
U
t
L
b la d e
Figure 2.13: Rounded edge geometry used by Hoerner (1965). This geometry with
′
Lblade /t = 6 has a drag coefficient with t as reference length of CD
= 0.64.
Hoerner (1965) (see also Blevins (1992)) found experimentally that a blade with
a rounded nose and a squared edged base, with the dimensions Lblade /t = 6 (see
′ = 0.64 for Re
4
figure 2.13) has a drag coefficient CD
Lblade > 10 . This geometry
is comparable to our rotor blade, except for the geometry of the trailing edge. The
ratio of the thickness and the blade length of a rotor blade of turbine meter 1 is
Lblade /t ≈ 20 and for turbine meter 2 the ratio is Lblade /t ≈ 8. The chamfered, sharp
edge reduces the drag coefficient, because the flow will not separate immediately at
the edge, which reduces the thickness of the wake. This is illustrated in figure 2.14.
u
n
in ,r e l
ro to r b la d e
A
f
m
u
w
w a k e
C S
o u t,r e l
w a k e
A
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 by
skin friction. The skin friction will be calculated separately. To determine the effect
of the skin friction compared to the pressure drag, the drag coefficient caused by laminar and turbulent boundary layers is now estimated by considering the rotor blade as
a flat plate. For laminar boundary layers the wall shear stress can be calculated using
Blasius’ numerical result (Schlichting (1979) and Appendix B). The drag coefficient,
28
2. Turbine flow meters in steady flow
′ , caused by the skin friction on both side of the blades is:
CD
′
CD,f
riction
=
2
R Lblade
τw dx
x=0
1
2
2 ρU t
1.328 Lblade
=p
ReLblade t
(2.21)
For turbulent boundary layers the drag coefficient is found empirically (Schlichting,
1979) to be:
−1
′
5
CD,f
riction = 0.148 ReLblade
Lblade
t
(2.22)
For the rotor blades considered in this chapter, the contribution of the skin friction
to the drag coefficient depends on the Reynolds number and whether the boundary
layers are laminar or turbulent. For the range of Reynolds numbers used in the present
experiments the contribution to the drag coefficient of the skin friction is typically
′ ≈ 0.05 for laminar boundary layers and C ′ ≈ 0.25 for turbulent boundary layers
CD
D
′ ≈ 0.03 for laminar boundary
for turbine meter 1. For turbine meter 2 we find CD
′
′ = 0.64 for
layers and CD ≈ 0.18 for turbulent boundary layers. As the total drag CD
the blade geometry with a blunt trailing edge (see figure 2.13 and Hoerner (1965)), we
expect that the contribution of the pressure drag will be in the order of 0.5. Assuming
that the wake has a thickness equal to the blade thickness, wwake = t, and that the
velocity in the wake is half the mainstream velocity, uwake = 12 uin , using equation
′ = 0.5. In case
2.20, we can calculate that the rotor blade has a drag coefficient CD
of the rotor blade with a chamfered trailing edge, we will also assume a wake with a
velocity uwake = 12 uin . The wake thickness, wwake will be tuned in order to match
′ for a two-dimensional model of the rotor blade. The
the measured values of CD
experiments 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 height hwt = 0.5 m and width wwt = 0.5 m
a two-dimensional wooden model of a single rotor blade is placed. The blade model
has a thickness, t, of 1.8 cm, a length, Lblade of 14.6 cm and a width, wblade of 48.9
cm. It has a rounded leading edge and a chamfered trailing edge (see figure 2.15).
The angle of the trailing edge is 45◦ .
The blade is connected to two balances with rods and ropes. The first balance
is a Mettler PW3000 with a range of 3 kg and measures the drag force, FD induced
by the flow around the blade. The second one is a Mettler PJ400 with a 1.5 kg
ranges and measures the lift force, FL . Both mass balances have an accuracy of 0.1
g. Measurements were carried out for Reynolds numbers, Re = uLblade /ν, based
on the blade length, ranging from ReLblade = 4 × 104 up to 3 × 105 at blade angles,
2.4. Wake behind the rotor blades
29
h
t
a
u
L
b la d e
w
w t
b la d e
w
w t
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).
′ = C Lblade = F / 1 ρu2 w
In figure 2.16 the drag coefficient, CD
D
D
blade t , is
t
2
plotted against the Reynolds number, Re = uLblade /ν for a blade angle, α = 0.3◦ .
′ , between 0.1 to 0.35, much lower
The measurements show a drag coefficient, CD
′
than CD = 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 boundary layers for the wind tunnel model. The contribution of the skin friction to the drag
coefficient is significant for turbulent boundary layers.
An other consequence of the asymmetric shape of the chamfered edge of the rotor
blade, is that at zero incidence, α = 0, the blade generates a lift force. This can be
seen in figure 2.17. This effect has not yet been included in the theory described in
this chapter, because we expect that the lift coefficient of a blade in a cascade strongly
deviates from a single blade in uniform flow as presented here.
In a closed wind tunnel cascade measurements are only possible at 0◦ incidence,
because the walls prevent deflection of the flow. For measurements at different angles
of incidence a special cascade wind tunnel should be used (Jonker, 1995). Measurements obtained for a five blade cascade with typical ratio of distance between the
blades and blade length, s/Lblade , of 0.55 indicated that the measured drag coeffi′ , values are close to the value obtained for a single blade.
cient, CD
2.4.2
Effect of wake on the rotation
The model described above is included in the momentum equation. The reduced velocity in the wake can be described with the displacement thickness, δ1,wake , and the
30
2. Turbine flow meters in steady flow
Hoerner (1965)
C’D=FD/(1/2 ρ u2in,rel wblade t)
0.6
0.5
0.4
0.3
0.2
turbulent
0.1
laminar
0
0
0.5
1
1.5
Re
2
2.5
3
5
x 10
L
blade
′
Figure 2.16: The drag coefficient, CD
, as a function of Reynolds number, ReLblade for flat
plate with round nose and 45◦ chamfered trailing edge measurements at an
angle of attack α = −0.3◦ . The arrow indicates the drag coefficient of 0.64
found in Hoerner (1965), the dashed line is an approximation for the part of
the drag coefficient in case of laminar boundary layers and the solid line is the
approximation for turbulent boundary layer.
0.5
blade
blade
0.2
0.1
0
L
C =F /(1/2 ρ u2
w
L
0.3
in,rel
)
0.4
L
−0.1
−0.2
−0.3
−3
−2
−1
0
α (o)
1
2
3
Figure 2.17: The lift coefficient, CL , is plotted at various angles of attack, α, for Reynolds
number, ReLblade > 3 105 . The dashed line is a linear fit through the data
points. We observe a net lift coefficient CL′ (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). Applying
mass conservation and using the displacement thickness, we find:
uout,x =
1
1−
nδ1,wake
2πr cos β
uin,x ,
(2.23)
where n is the number of blades of the rotor. Using both the displacement thickness, δ1,wake , the momentum thickness, δ2,wake , and equation 2.23 in the momentum
balance, the driving torque (equation 2.8) becomes:
Z rtip
u2in,x (tan β)r
n(δ1,wake + δ2,wake )
Td = −
ρ
dr
2 2πr −
nδ
cos β
rhub
1 − 2πr1,wake
cos β
Z rtip
+ 2π
ρuin,x ωr3 dr .
rhub
In the proposed model, in which the velocity in the wake, with a wake thickness
wwake , of half of the mainstream velocity, uwake = 12 uout , the displacement thickness
is δ1,wake = 12 wwake and the momentum thickness is δ2,wake = 14 wwake .
From the wind tunnel measurements described above, it is found that the drag
caused by the wake behind the blade is overestimated by using the drag coefficient
′ = 0.64. To account for this, the thickness of the wake can
in Hoerner (1965) of CD
be 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, the
drag coefficient of the rotor blade can be reduced to the values obtained from the
measurements. This will be applied in our calculations
Neglecting friction forces and assuming a uniform inflow, the deviation from the
ideal rotor speed caused by different drag coefficients, or different wake thickness
wwake , has been calculated. For the turbine flow meters 1 and 2 the effect of wake
thickness can be seen in table 2.2. In this approximation this effect is not dependent
on Reynolds number. We observe a significant effect of the drag on the deviation, E,
of the order of 2%.
2.5
Friction forces
Although turbine flow meters are designed to rotate with minimum friction, there are
several important friction forces that influence the rotation speed of the rotor. There
are two different kind of friction forces, the mechanical friction force and the friction
forces induced by the flow. Mechanical friction forces are the forces caused by the
bearings and the magnetic pick up placed on the meter. Flow induced friction consists
of the fluid drag on the blades and on the hub, the fluid friction at the tip clearance
32
2. Turbine flow meters in steady flow
thickness of the wake
0
1
2t
t
id
deviation, E = ω−ω
ωid × 100%
turbine meter 1 turbine meter 2
0
0
1.6
1.7
3.2
3.3
Table 2.2: The effect of the wake drag on the deviation of the rotation speed of the rotor from
ideal rotation for turbine flow meter 1 and 2, where t is the blade thickness.
and it includes the pressure drag due to the wake behind the blades discussed in the
preceding section. To approximate the friction forces on the rotor blades and the
hub, boundary layer theory has been used, neglecting centrifugal forces as well as
the radial velocity. In recent years numerical studies on turbine flow meters (Von
Lavante et al., 2003) show that the flow in the rotor has a complicated 3-dimensional
structure invoking secondary flows. It should be realised that the theory presented
here is a very simplified approximation of reality.
In the following sections the effect of these forces on the deviation from ideal
rotation will be investigated and discussed separately for both meters discussed in
2.2.3.
2.5.1
Boundary layer on rotor blades
Boundary layers are formed on the rotor blades as a result of friction. The boundary
layer thickness can be calculated using boundary layer theory and is included in the
momentum equation (equation 2.2). We assume that the cascade of rotor blades
can be described as row of rectangular channels with boundary layers at the top and
bottom of each channel. We neglect centrifugal forces and assume that there is no
radial velocity component. The rotor consists of n rectangular channels with a length
of Lblade (the length of the blade) and a width of hblade (the height of the blade). The
t
distance between two successive blades is 2πr
n − cos β . We consider two cases: the
case of a laminar boundary layer and the case of a turbulent boundary layer. The
displacement thickness, δ1,bl , the momentum thickness, δ2,bl , of the boundary layer
formed in this channel is calculated using the Von Kármán equation (2.17). For the
laminar case a third order polynomial is used to describe the velocity profile in the
boundary layer. For the turbulent case a 1/7th power law approximation is used. The
velocity between the blades is corrected for the displacement due to the growth of
the boundary layers in the channel. Using the mass conservation for incompressible
2.5. Friction forces
33
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 thickness, δ2,bl , (see Appendix B), for the boundary layer thickness at the end of the channel (the trailing edge of the blade), the equation for the driving torque, Td , becomes:
Td = −
Z
rtip
rhub
ρ
u2in,x (tan β)r
n(δ1,wake +2δ1,bl )
2πr cos β
2 ×
1−
Z rtip
n(δ1,wake + δ2,wake + 2(δ1,bl + δ2,bl ))
2πr −
ρuin,x ωr3 dr ,
dr + 2π
cos β
rhub
where δ1,wake is the displacement thickness caused by the wake and δ2,wake is the
momentum thickness caused by the wake (section 2.4.2). The rotation speed of the
rotor can now be found by determining iteratively at which rotational speed the total
torque in the equation of motion (2.13) is zero. This is determined numerically with
the secant method, a version of the Newton-Raphson method. In figure 2.18 the effect
of the boundary layers on the two different types of turbine flow meters for steady
incompressible flow with uniform inflow velocity and infinitesimally thin blades and
without other friction forces.
The laminar boundary layer causes the rotor to rotate faster, because the displacement thickness of the thicker laminar boundary layers. For the range of calculated
Reynolds 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, but
also 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 boundary
layer as a boundary layer on a long flat plate of a width w = 2πrhub − nt, where rhub
is the radius of the hub, n is the number of blades and t is the blade thickness. The
velocity outside the boundary layer will be assumed constant for simplicity reasons
and equal to the relative velocity:
urel
v
u
u
=t
U
1+
nt
2π cos βhub
!2
+ (ωrhub )2 ,
(2.25)
34
2. Turbine flow meters in steady flow
16
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
12
id
10
8
id
E = (ω − ω ) / ω * 100 (%)
14
6
4
2
0
−2 3
10
4
5
10
10
6
10
ReL
blade
(a) turbine meter 1
7
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
5
4
id
E = (ω − ω ) / ω * 100 (%)
6
id
3
2
1
0
−1 3
10
4
5
10
10
6
10
ReL
blade
(b) turbine meter 2
Figure 2.18: The deviation from ideal rotation caused by the boundary layers on the blades
for turbulent and laminar boundary layers, assuming a uniform inlet velocity
profile, versus Reynolds number, Re = U Lblade /ν.
where U is the velocity at the entrance of the turbine meter corrected for the displacement due to the boundary layers (see equation 2.18). Again, we assume that there is
no radial velocity and secondary flow. To determine the shear stress, τw , caused by
this boundary layer, two limits are considered. The first case is the upper limit for the
shear stress; the boundary layer starts at the entrance of the rotor. The flat plate has
a length of cosW
βhub , where W is the width of the rotor and βhub is the angle of the
rotor blades with the rotor axis at the hub of the rotor (see figures 2.1 and 2.2). The
second case is the lower limit; the boundary layer starts at the front end of the hub
2.5. Friction forces
35
and continues at the rotor. In this case the flat plate model of the flow has a length of
Lhub + cosW
βhub . The empirical expression for shear stress for a turbulent boundary in
a circular pipe, equation B.20, has been found to be a good approximation for the flat
plate (Schlichting, 1979). Using this equation and the equation for boundary layer
thickness for turbulent flow for a flat plate, the shear stress, τw , becomes:
9
1
1
5
τw = 0.0288 ρurel
ν 5 x− 5 .
(2.26)
The upper limit of the friction torque relative to the rotor axis due to the boundary
layers on the hub of the rotor, Tf r,hub = Ff r,hub rhub sin βhub , is:
Tf r,hub <wrhub sin βhub
Z
W
cos βhub
τw dx
0
1
5
W
cos βhub
4
W
Lhub +
cos βhub
4
9
5
=0.036 ρν urel wrhub sin βhub
(2.27)
5
,
and the lower limit is;
Tf r,hub >wrhub sin βhub
Z
Lhub + cosW
β
hub
τw dx
Lhub
1
5
9
5
=0.036 ρν urel wrhub sin βhub
"
5
4
5
− Lhub
# (2.28)
.
Using this in the equation of motion for the rotor (equation 2.13) and assuming
steady uniform flow and infinitesimally thin blades, while neglecting all other friction
forces, including the boundary layer on the rotor blades, the deviation from the ideal
rotation speed is computed. The result is plotted for different Reynolds numbers in
figure 2.19.
As expected the friction force on the hub slows down the rotor. For turbine meter
2, the larger flow meter, the effect is relatively small (at most 0.2%), while for turbine
meter 1, the effect can reach 1.5%. The ratio between the effect of the upper (equation
2.27) 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. This
imposes an additional drag force on the rotor. The force caused by the flow around the
tip 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 turbine
meter 1, the tip is enclosed in a slot (see figure 2.20).
36
2. Turbine flow meters in steady flow
0
E = (ω − ωid) / ωid * 100 (%)
−0.2
−0.4
−0.6
−0.8
−1
air at 1 bar, lower limit
air at 1 bar, upper limit
natural gas at 9 bar, lower limit
natural gas at 9 bar, upper limit
−1.2
−1.4 3
10
4
5
10
10
6
10
Re
L
blade
(a) turbine meter 1
−0.02
−0.06
−0.08
id
E= (ω − ω ) / ω * 100 (%)
−0.04
id
−0.1
−0.12
−0.14
air at 1 bar, lower limit
air at 1 bar, upper limit
natural gas at 9 bar, lower limit
natural gas at 9 bar, upper limit
−0.16
−0.18 3
10
4
5
10
10
6
10
ReL
blade
(b) turbine meter 2
Figure 2.19: The upper an lower limits of the deviations from ideal flow caused by the boundary layers on the hub, assuming a uniform inlet velocity profile, as a function of
the Reynolds number, Re = U Lblade /ν.
Thompson and Grey (1970) suggested that the tip clearance drag can be considered to be similar to the drag in a journal bearing. This results in friction torque
caused by the tip clearance, Ttc of
Ttc =
0.078
ρu2rel,rtip rtip Lblade tn .
2 Re0.43
tip
(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.
2.5. Friction forces
37
p ip e
p ip e
ro to r b la d e
ro to r b la d e
(a )
(b )
Figure 2.20: (a) shows an enclosed blade tip, (b) shows blade tip not enclosed
However, if vortex shedding occurs at the front edge of the tip and the blade
length is relatively small, the flow is no longer comparable to the flow in a journal
bearing. This makes it very difficult to give a reasonable prediction of the tip clearance drag. Therefore, no theory for the tip clearance is incorporated in the present
calculations. However, to understand the effect caused by tip friction calibration data
was obtained by Elster-Instromet for three turbine meters with identical geometries
except for the height of the tip clearance. These Reynolds curves and the calculated
Reynolds 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 mechanical friction
is caused by the bearing friction of the rotor. For a small meter the magnetic pick
up and the index counter can cause some additional friction, for larger meters this
part 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 to
determine the mechanical friction. Two different approaches were used to determine
the mechanical friction torque: dynamic and static friction measurements. These
measurements are discussed in more detail in section 4.2.3. The friction torque found
in the experiments are of the same order of magnitude as the data provided by the
manufacturer. For turbine meter 1 a mechanical friction torque of Tmech = 5.6×10−6
N m is assumed. For turbine meter 2 a torque of Tmech = 5.5×10−5 N m is assumed
based on the data of Elster-Instromet. Again, the equation of motion for the rotor was
solved for steady uniform flow with a rotor with infinitesimally thin blades and no
other friction forces than the mechanical friction torque. The deviation is shown in
figure 2.21.
The deviation caused by the mechanical friction does not depend on Reynolds
number, but depends on the volume flow and density, causing different behaviour at a
38
2. Turbine flow meters in steady flow
0
−2
id
−3
−4
id
E = (ω − ω ) / ω * 100 (%)
−1
−5
−6
−7
air at 1 bar
natural gas at 9 bar
−8
−9 3
10
4
5
10
10
6
10
ReL
blade
(a) turbine meter 1
−2
id
−4
−6
id
E = (ω − ω ) / ω * 100 (%)
0
−8
−10
−12
−14 3
10
air at 1 bar
natural gas at 9 bar
4
5
10
10
6
10
Re
L
blade
(b) turbine meter 2
Figure 2.21: The deviations from ideal rotation caused by the mechanical friction assuming
a uniform inlet velocity profile as a function of the Reynolds number, Re =
U Lblade /ν.
certain Reynolds number for air at 1 bar and natural gas at 9 bar. We see from figure
2.21 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 all friction
forces discussed in the previous sections and assuming that the flow entering the rotor is non-uniform, is compared to the calibration measurements of the two turbine
2.6. Prediction of the Reynolds number dependence in steady flow
39
meters; turbine meter 1 (Instromet SM-RI-X G250) and turbine meter 2 (more information about the meters can be found in table 2.1). The measured data is obtained
from the manufacturer of the turbine meter Elster-Instromet. The results of the calculations shown in the figures below are found assuming an inlet velocity profile based
on boundary layer theory with turbulent boundary layers. For the friction on the hub
the upper limit described in section 2.5.2 is used, while tip friction is ignored. For
the drag caused by either laminar or turbulent boundary layers the predictions for the
rotor blades are shown in the figures presented below. In the figures presented below
the prediction for both laminar as well as for turbulent boundary layers on the rotor
blade are shown.
2.6.1
Turbine meter 1
Typical calibration data is obtained for air at 1 bar (atmospheric pressure) and natural
gas at 9 bar 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 1 bar
and of natural gas at 9 bar and the dimensions given in table 2.1. Because it is not
known whether the boundary layers on the blades are either turbulent or laminar,
both situations are plotted in figure 2.22. The effect of the wake is calculated for a
′ = 0.5, based on the experiments of Hoerner (1965) as
pressure drag coefficient of CD
proposed in section 2.4, for a wake of thickness wwake = 21 t (this leads to a pressure
′ = 0.25) and for no wake (C ′ = 0).
drag coefficient of CD
D
′ = 0.5 predicts measurement errors, E, almost 4
The pressure drag coefficient CD
times lower than the measured data. To be able to obtain data similar to the measured
data an unrealistic high drag coefficient would be needed. It is unrealistic that the
model shows better agreement with the measured data using a drag coefficient much
higher than that of the blunt object measured by Hoerner (1965). Other reasons for
the observed high values of E have to be found. Deviation between theory and experiment could for example be due to the blade experiencing a nonzero lift coefficient at
a zero angle of attack (see section 2.4.1 and figure 2.17). This additional lift would
result in a higher rotor speed and accordingly a higher error, E. Another effect that
can account for the deviation between theory and experiment is the inflow velocity
profile. The measured inlet velocity profiles described in section 2.3.2 are used to
calculate the deviation between actual rotation speed and ideal rotation speed. In
figure 2.23 the deviation from ideal flow is plotted using the measured inlet velocity
profile in the calculations. The calculations have been carried out for both turbulent and laminar boundary layers on the rotor blades and a pressure drag coefficient
′ = 0.5. The actual velocity profile in the turbine meter at the inlet of the rotor
CD
induces a larger rotation speed, however, it cannot explain the difference between the
40
2. Turbine flow meters in steady flow
30
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
20
15
id
E = (ω − ω ) / ω * 100 (%)
25
measurements
id
10
5
C’ =0.5
C’DD=0.25
C’D=0
0
−5
−10 3
10
4
5
10
10
6
10
ReL
blade
Figure 2.22: The deviations from rotation versus Reynolds number, Re = U Lblade /ν for
turbine meter 1 in air at 1 bar and natural gas at 9 bar. The measured data
for air at 1 bar are indicated by circles • , the solid symbols represents the
data for a meter with standard blades, the open circles are for a blade with
a chamfered leading and trailing edge. The data indicated by triangles H are
for natural gas at 9 bar. The lines represent the calculated data, the results of
the calculations are for turbulent boundary layers as well as laminar boundary
layers 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 1 bar the standard
rotor was replaced by a rotor with blades, where the leading and the trailing edge are
both chamfered (figure 2.24(b)). These results are plotted in figure 2.22 as the open
dots. By changing the shapes of the blades, the shape of the error curve as function
of Reynolds changes. In the present case the effect of the shape of the leading edge
of the rotor blade is quite small.
To compare the shape of the measured curve with the shape predicted by the
model, we shifted the calculated data by 12.7% (see figure 2.25). The small deviations caused by the different blade profile cannot be explained using this global
model. The measured data resemble the prediction of the model with turbulent
boundary layers on the rotor blades much more than the model with laminar boundary
2.6. Prediction of the Reynolds number dependence in steady flow
41
30
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
20
15
id
E = (ω − ω ) / ω * 100 (%)
25
id
10
5
C’ =0.5
D
0
−5
−10 3
10
4
5
10
10
6
10
Re
L
blade
Figure 2.23: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν for
turbine meter 1 in air at 1 bar and natural gas at 9 bar. The symbols indicate
the results of the calculations using the measured velocity profile for laminar
boundary layers () and turbulent boundary layers (♦) on the rotor blades for
′
CD
= 0.5. The lines represent the calculated data using a turbulent boundary
layer model for the inlet velocity profile. The calculations are performed for turbulent boundary layers as well as laminar boundary layers on the rotor blades
′
at CD
= 0.5 (solid, dashed, dotted and dashed-dotted lines). The measured
data for air at 1 bar are indicated by circles •. The data indicated by triangles
H are for natural gas at 9 bar.
(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 a
chamfered leading edge used in the measurements
42
2. Turbine flow meters in steady flow
24
20
18
16
14
id
id
E = (ω − ω ) / ω * 100 (%)
22
12
10
8
6
4 3
10
4
5
10
10
6
10
Re
L
blade
Figure 2.25: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν
for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The calculated data
is shifted upwards by 12.7%. The measured data for air at 1 bar are indicated
by circles • , the solid symbols represents the data for a meter with standard
blades, the open circles are for rotor blades with a chamfered leading edge. The
data indicated by a triangle H have been measured in natural gas at 9 bar. The
lines represent the calculated data, the calculations are performed for turbulent
boundary layers (dotted: air at 1 bar; dash-dot: gas at 9 bar) as well as laminar
boundary layers (solid: air at 1 bar; dashed: gas at 9 bar) on the rotor blades.
layers over the entire range of Reynolds numbers considered. For the inlet velocity
profile a turbulent boundary model was used. Figure 2.23 shows that the measured
velocity profile changes the shape of this curve considerably. We conclude that the
manufacturer has used modifications of blade tip geometry (figure 2.20) in order to
compensate 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 of
the calculations for the model assuming either turbulent or laminar boundary layers
′ = 0.5,
on the blade. Again, this is done at different pressure drag coefficients, CD
0.25 and 0. The results are shown in figure 2.26. Again, a drag pressure coefficient
2.6. Prediction of the Reynolds number dependence in steady flow
43
30
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
20
15
id
E = (ω − ω ) / ω * 100 (%)
25
10
id
measurements
C’D=0.5
C’ =0.25
C’D=0
5
0
D
−5
−10 3
10
4
5
10
10
6
10
ReL
blade
Figure 2.26: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν
for turbine meter 2 in air at 1 bar and natural gas at 9 bar. The measured
data for air at 1 bar are indicated by circles • and for natural gas at 9 bar
by triangles H. The lines represent the calculated data, the calculations are
carried out for turbulent boundary layers as well as laminar boundary layers
on the rotor blades at three different drag coefficients (solid, dashed and dotted
lines).
′ = 0.5 results in deviations from ideal rotation considerably lower than the
of CD
measured data, as was the case for turbine meter 1. The deviation between the theory
and the experiments could be caused by the nonzero lift coefficient at a zero angle of
attack (figure 2.17). This additional lift can cause a higher error, E. A comparison
between the shape of the measured curve and the calculated curve is made by adding
6.5% over the whole Reynolds range (see figure 2.27). The deviation measured for
turbine 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 tip of
the rotor. As explained in section 2.2.3 a deviation from full fluid guidance reduces
the rotation speed and the error, E. The shape of the curve predicted by the model
agrees very well with the measured data assuming laminar boundary layers for the
air 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 measurement
conditions at the same Reynolds numbers. Since the measurement have been done
for two different set ups it is possible that the different conditions in the set up can
44
2. Turbine flow meters in steady flow
16
E = (ω − ωid) / ωid * 100 (%)
14
12
10
8
6
4
2
0
−2 3
10
4
5
10
10
6
10
Re
L
blade
Figure 2.27: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν
for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The calculated
data is shifted upwards by adding 6.5%. The measured data for air at 1 bar
are indicated by means of circles •. The data indicated by means of triangles
H are for natural gas at 9 bar. The line represents the calculated data, the
calculations are carried out for turbulent boundary layers (dotted: air at 1 bar;
dash-dot: gas at 9 bar) as well as laminar boundary layers (solid: air at 1 bar;
dashed: gas at 9 bar) on the rotor blades.
force the boundary layers to be either turbulent or laminar. No measurements were
done to ascertain the velocity profile at the inlet of the rotor for turbine meter 2. We
cannot determine more accurately the influence of the inflow velocity profile in this
case. However, we expect this would influence the deviation, E.
Comparison between turbine meter 1 with ”enclosed” blad tips and turbine meter
2 with normal blade tip clearance (figure 2.20) clearly illustrates the importance of the
tip clearance on the response of the flow meter. The Reynolds number dependence
of the measurement error, E, is much stronger for turbine meter 2 than for turbine
meter 1.
2.6. Prediction of the Reynolds number dependence in steady flow
diameter, D (m)
blade thickness, t (mm)
number of blades, n
rhub /D
rout /D
S/D
W/D
hblade /D = (rtip − rhub )/D
Lhub /D
45
0.280
3.0
24
0.359
0.500
2.707
0.136
0.138, 0.137, 0.136
1.532
Table 2.3: Dimensions of the turbine flow meters 3, 4 and 5 with different height of the tip
clearance, where rhub 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 and Lhub is the length of the hub in front of the
rotor. Except for the blade thickness t and the number of blades n, all values are
made 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 clearance. Because no satisfactory model has been found to describe the friction caused
by the flow through the gap, this effect is not included in our model. To estimate
the effect of different tip clearance, our model is compared to a series of turbine meters with the same geometry, but with different height of the tip clearance, htc =
rout − rtip = 1, 2, 4 mm (see figure 2.28). As for turbine meter 2 the tip is not enclosed in a wall cavity (figure 2.20). The dimensions of the meter are given in tabel
2.3.
p ip e
r
o u t
r
tip
h
tc
ro to r b la d e
Figure 2.28: Tip clearance of a blade that is not enclosed.
46
2. Turbine flow meters in steady flow
The calibration data for all three turbine meters (turbine meter 3, 4 and 5) are
plotted in figure 2.29. The measurements show a decrease in E for increasing tip
clearance height, htc .
The theoretical results are shown in figure 2.30, assuming an inlet velocity profile
′ = 0.5
based on turbulent boundary layers, the drag coefficient is taken to be CD
and using the upper limit of the hub friction. Although there is no model for tip
clearance drag, there is still an effect, because the height of the blade decreases for
increasing gap height, htc . For larger blade heights less of the blade is rotating in the
boundary layer at the outer wall of the pipe, this causes the increase of rotation speed
of the rotor for larger tip clearance as can be seen in figure 2.30. If the effect of tip
clearance drag suggested by Thompson and Grey (1970), equation 2.29, is taken into
account, the results do not change significantly. Because the mechanical friction of
this particular turbine meter has not been measured, it was assumed to be equal to that
of turbine 2. It is however clear from the figures that in reality the mechanical friction
is 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 describe the effect of tip clearance drag correctly. The results of the experiments shown
in figure 2.29 display exactly the opposite behaviour of the results obtained with our
model shown in figure 2.30. As the tip gap increases our model predicts an increasing
rotational 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 and of
changes in fluid properties, a theoretical model of the behaviour of the turbine flow
meter in steady flow is necessary. The flow in a turbine meter is 3-dimensional and
very complicated. We consider a very simplified model. The model presented in this
chapter makes it possible to calculate for a chosen geometry the Reynolds number
dependence 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 measurements 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 friction
and a wake displacement effect. The wake thickness can be tuned to match the drag
coefficient 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 account
for the high rotation speed of the rotor compared to ideal rotation by modifying this
wake thickness only. A possible explanation is that the flow also generates a lift force
on the rotor blades at zero angle of attack (figure 2.17). The lift causes the rotor to
rotate faster. A lift force on a single rotor was measured in wind tunnel experiments.
2.7. Conclusions
47
18
E = (ω − ωid) / ωid * 100 (%)
16
14
12
10
8
tc=1 mm; air at 1 bar
tc=1 mm; natural gas at 9 bar
tc=2 mm; air at 1 bar
tc=2 mm; natural gas at 9 bar
tc=4 mm; air at 1 bar
tc=4 mm; natural gas at 9 bar
6
4
2
0 3
10
4
10
5
10
Re
6
10
7
10
L
blade
Figure 2.29: The measurement deviation from ideal rotation for turbine meters with tip clearance height of 1, 2 and 4 mm versus Reynolds number.
18
air at 1 bar, turbulent
air at 1 bar, laminar
natural gas at 9 bar, turbulent
natural gas at 9 bar, laminar
E = (ω − ωid) / ωid * 100 (%)
16
14
12
10
8
6
4
2
0 3
10
4
10
5
10
ReL
6
10
7
10
blade
Figure 2.30: Calculated deviation of ideal rotation for turbine meters with tip clearance
height, htc = 1 mm (solid line), htc = 2 mm (dashed line) and htc = 4 mm
(dotted line). Calculation are for laminar as well as turbulent boundary layers
on the rotor blades and the solid symbols represent the data using the equation
of tip clearance drag suggested by Thompson and Grey (1970), equation 2.29.
′
The drag coefficient used is CD
= 0.5.
48
2. Turbine flow meters in steady flow
Measurements on a cascade are needed to determine this lift force for a turbine meter
type of rotor.
The present model gives an adequate prediction of the shape of the Reynolds
number dependency of the rotor response. However, the effect of changing the shape
of 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 strong
influence on the Reynolds number dependence of the response of the rotor.
3
Response of the turbine flow meter on
pulsations with main flow
3.1
Introduction
Gas turbine flow meters can reach high accuracy, generally of the order of 0.2%. This
accuracy can only be attained for optimal flow conditions. Acoustic perturbations can
induce significant systematic errors. A theoretical prediction of the error would allow
a correction in the volume flow measurement.
In recent years research has been carried out to determine the pulsation error
during 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 the
real volume flow. This method can only be used if the amplitude of the pulsations can
still be detected in the turbine signal. As the amplitude of the pulsations in the turbine meter signal decreases rapidly with increasing frequency, it is difficult to predict
the real flow for high frequency pulsations. Another tool was developed described
by Cheesewright et al. (1996), called the ’Watchdog System’. This system also uses
the equation of motion of the rotor (equation 3.5), but now an accelerometer is used
to measure the flow noise of a valve or bend. Watchdog is designed for pulsation
frequencies less than 2 Hz. We actually focus on the behaviour of the rotor at high
frequencies for which the rotor inertia has integrated fluctuations in rotor speed. The
errors we consider are due to non-linearities.
Assuming quasi-steady, incompressible flow and neglecting friction forces a relationship can be found between the velocity pulsations and the measurement error
(Dijstelbergen, 1966). Experiments have been carried out in the past by Lee 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 pulsations. To explore the limits of the validity of this theory, a set up was build at the
Eindhoven University of Technology.
In our experiments care was taken to determine accurately the amplitude of the
velocity fluctuations at the rotor. This was found to be a limitation in the experiments
reported in literature. With the more accurate determination of the acoustic field
it is possible to detect small deviations from the basic theory. With this set up it
was possible to measure the influence on the flow meter response of the acoustic
perturbation with velocity amplitudes from 2% of the main flow velocity up to twice
the main flow velocity for frequencies from a few Hz up to 730 Hz. In this chapter
the basic theory is discussed, after this the set up is described and the experimental
procedures are discussed. We then show the results of the measurements and discuss
these results.
3.2
3.2.1
Theoretical modelling
A basic quasi-steady model: A 2-dimensional quasi-steady model for a
rotor with infinitesimally thin blades in incompressible flow
If the rotor is modelled as a 2-dimensional cascade of infinitesimally thin blades in an
incompressible, frictionless, steady flow, the integral mass and momentum equation
applied to a fixed control surface CS with an outer normal n (equations 2.1 and 2.2)
reduce to
ZZ
v · ndS = 0 ,
ρ0
(3.1)
CS
ZZ
(3.2)
ρ0
v (v · n) dS = F bf ,
CS
where ρ0 is the fluid density, v is the velocity vector and F bf are the forces imposed
on the fluid by the turbine blades. The control volume, CS, is chosen as shown in
figure 3.1 and we assume that there is no pressure drop over the cascade. Because
infinitesimally thin blades and frictionless flow are assumed, the surface area of inflow is equal to the surface area of outflow. It follows from equation 3.1 that the axial
component 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 momentum and that the flow leaves the rotor with a velocity aligned with the blades (see
fig. 3.1). This is a realistic assumption if the chord length of the blades, Lblade , is
large compared to the distances between the blades, s, i.e. for cascades the ratio,
3.2. Theoretical modelling
51
w r
W
b
u
u
in
u
re l
b
w r
u
o u t,x =
o u t,y
in
n
re l
t
n
u
u
L
C S
b la d e
y
x
Figure 3.1: Two-dimensional representation of the flow entering and leaving the rotor modelled as a cascade
s/Lblade , should be smaller than 0.7 (Weinig, 1964). By considering the flow in a
reference frame attached to the rotor, this implies that :
tan β =
ωr + uout,y
.
uout,x
(3.3)
We assume further that there is no swirl in the incoming flow, so that uin,y = 0 and
that the inflow is uniform. The momentum equation in the y-direction becomes
ρ0 Auin (uin tan βav − ωR) = Fbf,y ,
(3.4)
where ω is the angular rotation velocity of the rotor, A is the cross-sectional area of
the rotor, βav is the average blade angle and R is the root-mean-square q
radius of the
inner and outer radius of the meter, rin and rout respectively, i.e. R =
2 +r 2 )
(rin
out
.
2
52
3. Response of the turbine flow meter on pulsations with main flow
The force exerted on the fluid by the blade, Fbf,y , is equal and opposite to the force
exerted by the fluid on the blade, Ff b,y . The fluid induces a torque on the rotor,
Tf b = Ff b,y R, accelerating the rotor. Using the equation of motion of the rotor, we
get:
dω
= ρ0 Auin (uin tan βav − ωR)R − Tf ,
(3.5)
dt
where Irotor is the moment of inertia of the rotor and Tf is torque on the rotor caused
by the friction forces.
We assume periodic pulsations u′in around an average velocity ūin so that uin =
T
ūin + u′in . We neglect the friction torque, ρu2 fR3 ≪ 1. We assume that the rotation
in
of the rotor is constant in spite of the pulsations. The integration time of the rotor
is much longer than the period of the imposed acoustic pulsations. If the torque is
averaged over one period, equation 3.5 reduces to:
Z
1 T
ρ0 (ūin + u′in )A (ūin + u′in ) tan βav − ωR Rdt =
T 0
i
h
)
tan
β
−
ū
ωR
= 0,
(3.6)
ρ0 AR (ū2in + u′2
av
in
in
Irotor
with T is the period of the pulsations. From this equation the angular rotation velocity, ω̄, caused by pulsating flow is obtained
!
u′2
ūin tan βav
in
1+ 2
ω̄ =
.
(3.7)
R
ūin
βav
for the ideal angular rotation velocity
Using equation 1.3, i.e. ωid = ūin tan
R
without pulsation, the error caused by periodic pulsations becomes:
!
u′2
ω̄ − ωid
in
.
(3.8)
(Epuls )id =
=
ωid
ū2in
This means that sinusoidal pulsations, uin = ūin + |u′in | sin(ωt), induce a systematic
error of
1 |u′in | 2
(Epuls )id =
.
(3.9)
2 ūin
3.2.2
Practical definition of pulsation error
In the previous section we defined a deviation (Epuls )id between angular velocity, ω̄,
for steady rotation and the ideal angular rotation velocity, ωid , in absence of pulsations:
ω̄ − ωid
(Epuls )id =
,
(3.10)
ωid
3.3. Experimental set up
53
βav
. In experiments we use the steady angular velocity ω0 as
where ωid = ūin tan
R
reference instead of ωid , hence:
Epuls =
ω̄ − 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 the
influence of a constant mechanical friction torque, T̄mech on an ideal rotor. Using
equation 3.6 we find in absence of pulsations:
ω0 = ωid −
T̄mech
,
ρ0 AR2 ūin
(3.12)
while due to pulsations we would reach a steady rotation of angular velocity:
!
u′2
ūin tan β
T̄mech
1 + 2in −
.
(3.13)
ω̄ =
R
ρ0 AR2 ūin
ūin
Hence, we would predict a pulsation error, (Epuls )exp :
(Epuls )exp =
ωid
ω0
u′2
in
,
ū2in
(3.14)
u′2
corresponding to (Epuls )id = ū2in multiplied by a factor ωωid0 .
in
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 the
flow at high Strouhal numbers. Our aim is to provide quantitative information about
this Strouhal number dependence.
3.3
Experimental set up
A dedicated set up has been built at Eindhoven University of Technology to study the
influence of pulsations on gas turbine meters. A high pressure reservoir with dry air at
60 bar (dew point -40◦ C) is connected to a test pipe of 0.10 m diameter, and a length
of 3.2 m. At the open end of this pipe a turbine flow meter (Instromet type SM-RI-X
G250) is placed. The flow through the turbine flow meter is controlled by means
of a valve placed at the upstream end of the test pipe. By adjusting this valve, the
critical pressure at the valve is reached, resulting in a velocity, u∗ , at the valve equal
to the local speed of sound, c∗ . We have a Mach number of unity, M = u∗ /c∗ = 1
and a so-called ”choked” flow. This provides a constant mass flow, independent of
54
3. Response of the turbine flow meter on pulsations with main flow
A
a irre s e rv o ir
c h o k e d
v a lv e
C
F
s ire n
B
p 1 p 2 p 3 p 4 p 5 p 6
D
te m p e ra tu re
p 7 p 8
G
p u ls e
sh a p e r
8 x
h o tw ire
m e a su re m e n t
h w 1
h w 2
E
s ig n a l
g e n e ra to r
trig g e r
(8 )
(4 )
S & H
+
filte r
Figure 3.2: Experimental set up: A high pressure reservoir of dry air (A) is connected with
a pipe (B) to a turbine meter (D). The flow is being controlled by an adjustable
valve (C) creating choked flow with constant mass flow. Pulsations can be induced by a loudspeaker (E) or a siren (F). The pulsations are measured with
six pressure transducers (p1,p2,p3,p4,p5 and p6) along in the pipe (B) and two
pressure transducers (p7 and p8) placed within the turbine meter (D). Velocity
pulsations can be measured with two hot wires (hw1 and hw2) placed within the
turbine meter (D). The rotation of the rotor of the turbine meter is being measured
by a probe detecting the passing of a rotor blade (G).
perturbations in the flow downstream of the valve. The conditions of the reservoir,
p0 and T0 , and the valve opening determine the mass flow.
Pulsations in the test pipe downstream of the choked valve can be induced by
using a loudspeaker placed at the downstream open end of the set up or by means
of a siren placed downstream of the valve. The loudspeaker (SP-250P) is controlled
using a signal generator (Yokogawa FG120) driving a power amplifier (AIM WPA
301A). The siren is described by Peters (1993). The siren has a frequency range from
a 10 Hz up to 1000 Hz. A bypass allows variations in the ratio, uac /u0 , of acoustic
velocity, uac , and the main flow velocity, u0 . The siren is a much more efficient
sound source than the loudspeaker, by tuning it to the resonance frequencies of the
set up, the ratio of acoustic to main flow velocities, uac /u0 , can reach values up to
2. Between the siren and the valve a volume is placed, a pipe with a length 1.22 m
and a internal diameter of 0.21 m. Except for the core of the pipe with a diameter of
0.05 m, this pipe is filled with porous material (Achiobouw acoustic foam D80). To
avoid chocking at the siren, the opening of the bypass was increased, while the siren
3.3. Experimental set up
pressure transducer in pipe
1
2
3
4
5
6
pressure transducer in turbine meter
7
8
55
distance
-1.566 m
-1.212 m
-0.960 m
-0.400 m
-0.265 m
-0.205 m
distance
0.07775 m
0.1075 m
Table 3.1: Position of the pressure transducer placed in the set up. The distances are measured from the upstream end of the turbine meter, where the positive direction is
the flow direction.
was turned off, up to the point at which changes in volume flow could no longer be
observed. Only measurements using a significantly larger opening of the bypass than
this critical point are performed. Goog agreement between the measurements using
the 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-electric
gauges placed flush at the pipe wall. Six pressure transducers (three Kistler type 7031
and three PCB type 116A) are placed in the pipe upstream of the turbine meter each at
randomly chosen distances (see table 3.1). Two other pressure transducers (PCB type
116A) are placed within the turbine meter, 0.010 m and 0.040 m upstream of the rotor of the turbine meter (see figure 3.3). The signals from the pressure transducers are
amplified using charge amplifiers (Kistler type 5011). They are acquired by means of
a 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. The pressure
transducer and charge amplifier combinations are calibrated in a different set up. In
this calibration set up the transducer is placed next to a reference microphone flush in
a closed end wall of a 1.0 m long pipe (diameter 0.07 m). Plane waves are generated
by a loudspeaker placed at the opposite end of the pipe. All pressure transducers are
calibrated against the reference pressure transducer for frequencies between 24 Hz
and 730 Hz, i.e. the frequencies used in our experiments. The acoustic velocity of
the pulsations can also be measured using two hot wire anemometers (Dantec type
55P11 wire diameter 5 µm with 55H20 support) placed 0.010 m upstream of the turbine meter (see figure 3.3). Accurate measurements of the amplitude of the velocity
pulsations are only possible if the ratio between the acoustic velocity amplitude and
56
3. Response of the turbine flow meter on pulsations with main flow
p re s s u re tra n s d u c e r
h o tw ire a n e m o m e te rs
flo w s tra ig h tn e r
p re s s u re tra n s d u c e rs
ro to r
h o tw ire a n e m o m e te r
(a)
(b)
Figure 3.3: The placement of the pressure transducers in the turbine meter. (a) shows a
schematic, 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 hot
wires are placed at the same distance upstream from the rotor (1 cm), equally
distributed around the perimeter of the meter.
the main flow velocity is small enough to avoid flow reversal, uac /u0 < 1. The hot
wire makes no distinction between forward and reversed flow. The signals of the hot
wire anemometers are processed with a constant-temperature anemometer module
(Streamline 90n10) in combination with dedicated Dantec application software. The
anemometer can follow velocity fluctuations up to 50 kHz. The signals are recorded
on a PC in the same way as the signals of the pressure transducers. The hot wire
anemometers are calibrated against a Betz water micromanometer (± 1 P a) in a separate free-jet set up in the velocity range 2 m/s to 40 m/s. The output is fitted using
a power law description. This results in accuracies of about 1% for velocities above
8 m/s and 5% for velocities from 2 to 8 m/s.
The time-averaged volume flow can be measured using the turbine flow meter (Instromet type SM-RI-X G250), using the calibration data provided by Elster-Instromet
for normal flow conditions. In the absence of pulsations the volume flow measured
has an accuracy of 0.2 % in the range of 6 × 10−3 m3 /s to 1 × 10−1 m3 /s. The
rotation of the rotor of the turbine meter is detected by means of a so-called ”reprox
probe”, a magnetic pickup generating an inductive pulse when a rotor blade passes
the probe. These pulses are converted to electronic pulses and then modified into
proper TTL pulses by means of the signal generator. The time interval between the
TTL pulses is registered using a counter board (PCI 6250 NI), inserted in a PC, with
an accuracy of 50 ns. These intervals are converted to the rotation period of the rotor by multiplying by the number of rotor blades (n=16). Due to small differences
in 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 the
location of the rotor
When the loudspeaker or the siren is turned on, velocity pulsations are generated. The
velocity in the set up can be described by the average main flow, ūin , and a periodic
fluctuating part, u′in . To investigate the effect of the acoustic perturbations on the flow
measurements of the turbine meter, it is necessary to determine the velocity pulsations
at the position of the rotor. It is impossible to measure the velocity pulsations exactly
at the rotor. The measured data has to be extrapolated to the rotor position. By
using the measured pressure fluctuations obtained by the microphones, the acoustic
velocity at the rotor is determined by using an acoustic model.
3.4.1
Acoustic model
p ip e
p
tu rb in e flo w m e te r
p
1
-
1
+
p
p
2
-
2
p
+
p
3
3
+
-
Figure 3.4: A schematic illustration of the acoustic model used to determine the amplitude of
the velocity pulsations at the position of the rotor.
For the acoustic model the test pipe including the turbine meter is divided into
three parts with different cross-sectional areas (see figure 3.4). The first part is the
pipe leading to the turbine meter with a diameter D = 0.10 m. The pipe has a crosssectional area of 8.4 × 10−3 m2 . As can be seen in figure 3.3 the core of the turbine
meter has a complicated shape around the rotor. In the acoustical model, the turbine
meter will be described as two cylindrical parts changing abruptly in cross-sectional
areas. The first part of the turbine meter is the front part of the flow straightener and
has a length of 0.037 m and a cross-sectional area of 7.3 10−3 m2 . The second part is
the main part of the turbine meter and has a smaller cross-sectional area of 3.8 10−3
58
3. Response of the turbine flow meter on pulsations with main flow
m2 . It is assumed that the acoustic field can be described within each segment as
plane 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 the
one-dimensional equation:
+
−
i(2πf t−kj x)
i(2πf t+kj
p′j (x, t) = p̂j (x)ei2πf t = p+
+ p−
j e
j e
x)
, j = 1, 2, 3(3.15)
with p̂j the complex amplitude and f the radial frequency. In this case of an uniform
main flow, kj+ and kj− are the complex wave numbers of the waves travelling in
positive and negative direction, respectively. The wave numbers are defined as:
kj+ =
2πf /c0
+ (1 − i)αd ,
1+M
kj− =
2πf /c0
+ (1 − i)αd .
1−M
(3.16)
The imaginary part of the wave number represents the damping coefficient caused by
viscous-thermal effects. In a quiescent flow in smooth cylindrical pipes, damping of
plane waves by viscous-thermal damping can be described by a damping coefficient,
αd (Kirchhoff (1868), Tijdeman (1975), Pierce (1989))
γ−1
1 p
νπf 1 + √
αd =
,
(3.17)
r j c0
Pr
where rj is the radius of pipe segment j, ν is the kinematic viscosity, γ is the Poisson’s ratio and Pr is the Prandtl number. For air at room temperature and atmospheric
pressure 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 good
approximation of the effect of
qdamping for frequencies such that the acoustical visν
+
cous boundary layer, δ =
πf , is thinner than the viscous sublayer 10δ with
q
δ + = ν τρw (Peters (1993), Ronneberger and Ahrens (1977) and Allam and Åbom
(2006)). If we use the δ + for smooth cylindrical pipes, we find that this is valid for
our experiments.
At the abrupt transitions in cross-section the integral formulation of the conservation of mass flow, m′ , and total enthalpy, B ′ are used for compressible potential
flow (Hofmans, 1998);
m′1 = m′2 ,
B1′ = B2′ ,
Aj + −ikj+ x
ikj− x
m′j =
pj e
(1 + Mj ) − p−
e
(1
−
M
)
,
j
j
c0
1 + −ikj+ x
ikj− x
pj e
(1 + Mj ) + p−
e
(1
−
M
)
.
Bj′ =
j
j
ρ0
(3.18)
3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 59
By introducing a matrix M and a vector [pm ] with the pressures measured at
the microphones, the solution of the system of the equations 3.18 and the equations
3.16 at the positions of the microphones can be computed. The system of equation
becomes:
+
−1 T
pj
T
M · [pm ]
(3.19)
− = M M
pj
From this system of equation the least-square solution of the plane wave amplitudes
−
p+
j and pj is determined. Using
+
uac =
−
−ik3 xr − p− eik3 xr
p+
3e
3
,
ρ 0 c0
(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 being
used during post-processing. With synchronous detection, it is possible to measure
the 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 (2πf t). When the measurements are carried out by using a loudspeaker
to induce pulsations, the signal driving the loudspeaker is used as reference. When
the siren is used to induce pulsations, one of the pressure transducers is filtered out
digitally using a second order band-pass filter to produce a sinusoidal reference signal. From the sine wave reference signal a cosine wave signal is obtained by shifting
the phase by π/2. The Hilbert transform routine of Matlab is used to obtain the
shifted reference signal. The transducer signals are multiplied by the sine reference
signal and integrated over a integer number of oscillation periods to extract the amplitude of the sin(2πf t) component of the signal. The same procedure is repeated
for the cosine reference signal to obtain the amplitude of the cos(2πf t) component
of 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 acoustic
velocity several approaches are used. The pressure transducers placed in the turbine
meter are placed close to the rotor. The rotation of the rotor, the wake of the guiding
vanes of the flow straightener and the abrupt transitions in cross-section can cause interference on these pressure measurements. This can be investigated by excluding the
two microphones placed within the turbine meter. In figure 3.5 an example is given
60
3. Response of the turbine flow meter on pulsations with main flow
400
300
200
100
position
rotor
0
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbine meter (m)
(a) all pressure transducers
pressure amplitude (Pa)
pressure amplitude (Pa)
of an experiment in which pulsations were induced at 164 Hz. The figure shows the
pressure amplitude, the added upstream and downstream travelling pressure waves.
It can be found that the difference between these two models is small, in the order
of a few percent in velocity amplitude, depending on frequency and standing wave
pattern.
400
300
200
position
rotor
100
0
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbine meter (m)
(b) pressure transducers in the turbine meter
are excluded
Figure 3.5: The pressure amplitude of the standing wave in the set up during a measurement
at a frequency of f = 164 Hz with a mainstream velocity in the pipe of u0 = 2
m/s. The dots represent the measured pressure amplitude of the pressure transducers. Figure (a) shows a example of a measurement were all eight pressure
transducers in the set up are used, (b) shows the standing wave predicted when
the two pressure transducers in the turbine meter are not used. The different lines
indicate the three different parts of the acoustic model (figure 3.4).
The accuracy of the acoustic model depends on the position of the pressure nodes
of the standing wave. If the pressure node is located around the rotor position, small
deviations in the pressure wave induce small deviations in the velocity amplitude,
because the velocity is rather uniform around a pressure node. When the rotor is
close to a pressure antinode large errors in velocity amplitude can be induced by
small deviations. Measurements are only considered when the acoustical velocity
can be determined accurately. In figure 3.6 an example is given of a measurement
at 362 Hz, for which the position of the rotor is close to a pressure maximum. The
results of this experiment were therefore not used in our analysis.
To illustrate this, the velocity amplitude was calculated at the front of the rotor
and at the back of the rotor (the rotor has a width of 2 cm). In the case shown in figure
3.6(a), where the rotor position is around a pressure antinode, the velocity amplitude
changes over the width of the rotor with 28%, while in the figure 3.6(b) the velocity
amplitude changes with 10%. Besides the location of the rotor position in reference to
the standing wave, the frequency also plays an important roll. In figure 3.7 examples
3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 61
400
300
position
rotor
200
100
0
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbine meter (m)
(a) rotor position around pressure maximum
pressure amplitude (Pa)
pressure amplitude (Pa)
400
300
200
position
rotor
100
0
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbine meter (m)
(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 a
frequency of f = 362 Hz with a mainstream velocity in the pipe of u0 = 2 m/s.
The dots represent the measured pressure amplitude of the pressure transducers.
Figure (a) shows a example of a measurement where the rotor is located close to
a pressure antinode, (b) shows a measurement at the same frequency and main
stream velocity with the rotor not as close to the pressure maximum. The different
lines indicate the three different parts of the acoustic model (figure 3.4).
of a measurement at 24 Hz and a measurement at 730 Hz are shown. If we evaluate
the accuracy as mentioned above, for 24 Hz the velocity amplitude changes over the
rotor with less than 0.1%, while at 730 Hz this is about 3 percent. In this example
(730 Hz) this change in velocity amplitude is still relatively small, because the rotor
is positioned at a pressure node.
To verify the velocity amplitude found with the acoustical model further, the velocity amplitude is measured 1 cm upstream of rotor by means of two hot wires
placed at different positions. Two hot wires were used to account for the complicated
flow profile behind the blades of the flow straightener. The local relative velocity
pulsations for |u′ |/u0 < 1, can be compared with the relative velocity amplitude
calculated with the acoustical model based on the pressure measurements. The measurements of the velocity amplitude with the hot wires are within 10% in agreement
with the acoustical model for velocities higher than 2 m/s. Below 2 m/s the calibration of the hot wire is problematic. We will discuss the hot wire measurements
further in section 3.4.4.
The siren generated block pulses in volume flow, which drives many harmonics
of the fundamental frequency. But by using the siren only at resonance frequencies
of the pipe, the resonant frequency dominates over other frequencies. In that case
we obtain an almost harmonic perturbation. Some overtones will, however, still be
present. Using equation 3.9 it is expected that the contributions of the different har-
62
3. Response of the turbine flow meter on pulsations with main flow
300
400
300
200
100
position
rotor
0
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbinemeter (m)
(a) 24 Hz
pressure amplitude (Pa)
pressureamplitude (Pa)
500
250
200
150
100
position
rotor
50
−2
−1.5
−1
−0.5
0
0.5
distance from upstream end of turbine meter (m)
(b) 730 Hz
Figure 3.7: The pressure amplitude of the standing wave in the set up. The dots represent
the measured pressure amplitude of the pressure transducers. Figure (a) shows
a example of a measurement at 24 Hz, (b) shows a measurement at 730 Hz.
Both measurements are carried out at a mainstream velocity u0 = 2 m/s. The
different lines indicate the three different parts of the acoustic model (figure 3.4).
monics add quadratically to the error:
#
"
′ 2 ′ 2
1
|u2 |
|u3 |
|u′1 | 2
Epuls =
+
+
+ .....
2
uin
uin
uin
(3.21)
where the subscripts indicate the different harmonics.
This is checked by inducing pulsations using the loudspeaker and the siren simultaneously at different frequencies. As is shown in table 3.2 the measurement
error caused by the frequencies separately accumulates, Eadd , within measurement
accuracy to the measurement error caused by the two frequencies simultaneously,
Esim . When the difference in frequency is small the induced acoustical velocity will
display low frequency beats which the siren can follow. This produces the type of
signal shown in figure 3.8. As the frequency obtained by using the siren displays
some drift in time, we observe some time dependence in the frequency of the beats
for simultaneous measurements with two frequencies close together. As shown in
figure 3.8 at t = 20 seconds the loudspeaker is turned on and the turbine meter starts
to measure a higher velocity. After another 20 seconds the siren is turned on, the
measurement error becomes larger and starts oscillating. This is by the beats. Taking
the time average of the error during the beats we still find that Eadd ⋍ Esim (table
3.2).
For signals in which other frequencies are present, the contribution from each
frequency can be added to predict the total error. As the error depends quadratically
on the amplitude the harmonics with higher amplitude will dominate.
3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 63
|u′ |/u0
f = 24Hz
0.21
0.21
0.21
f = 164Hz
0.09
0.09
0.09
f = 164Hz
0.26
0.26
f = 164Hz
0.18
0.18
|u′ |/u0
f = 164Hz
0.19
0.33
0.57
f = 367Hz
0.07
0.12
0.16
f = 166Hz
0.18
0.25
f ≈ 164Hz
0.28
0.33
Esim −Eadd
Esim
Eadd
Esim
0.032
0.074
0.227
0.033
0.074
0.228
0.12 %
-0.18 %
0.64 %
0.005
0.011
0.015
0.005
0.010
0.016
-3.06 %
-2.73 %
1.04 %
0.036
0.049
0.040
0.048
6.17 %
-1.07 %
0.042
0.052
0.041
0.056
-2.32 %
7.03 %
× 100%
Table 3.2: Measurements with pulsations at two frequencies. Eadd is the added measurement
error of these frequencies separately and Esim is the measurement error for measurement for both pulsations simultaneously. All measurements are carried out at
a mainstream velocity of u0 = 2 m/s.
64
3. Response of the turbine flow meter on pulsations with main flow
2 .1 5
s ire n o ff
2 .1
v e lo c ity ( m /s )
2 .0 5
lo u d s p e a k e r o ff
2
1 .9 5
s ire n o n
lo u d s p e a k e r o n
1 .9
0
5 0
1 0 0
tim e ( s )
1 5 0
2 0 0
2 5 0
Figure 3.8: Mainstream velocity of the flow measured with the turbine meter. After 20 seconds the loudspeaker is turned on at a frequency of 164 Hz, after another 20
seconds 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 transducers
is not practical for industrial use of turbine flow meters. For this reason the options
to measure the velocity amplitude by means of a hot wire or two pressure transducers
embedded in the turbine meter has been investigated.
A hot wire determines the local velocity as well as velocity fluctuations, in the
set up two hot wires are placed. Both hot wires are placed 1 cm upstream of the
rotor and 1 cm from the pipe wall in the flow. One hot wire was placed in the wake
of 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 flow
profile around the flow straightener. The mainstream velocity measured locally with
the hot wires are higher, than the mean velocity measured by the turbine meter. This
is caused by hot wire measurements being local measurement, and the flow profile in
the annulus is not uniform. The local flow velocity can be higher or lower than the
mean velocity depending on the position of the hot wire. As expected the average
velocity in the wake of the vane is lower than the velocity measured between the
vanes of the flow straightener. The measured amplitude of the velocity pulsations for
high 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 measures
systematically a higher velocity amplitude than the other hot wire. This can probably
be 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 approximation
using only pressure transducers at two positions in the turbine meter close to the
rotor. By using the excitation frequency and considering only plane waves, we can
determine the amplitude of the velocity pulsation. Using the linearised momentum
equation for harmonic perturbations it follows that:
uac =
∆p′
,
2ρ0 πf Lm
(3.22)
with ∆p′ the pressure difference and Lm the distance between the two microphones.
Taking compressibility into account this becomes:
∆p′
∆p′ −i π2
2
e
(1 − M ) + M
uac = ,
(3.23)
2ρ0 πf Lm
ρ 0 c0 where M= u0 /c0 is the Mach number. The applicability of this method is very dependent on frequency and standing wave pattern, comparable to the situation shown
in figure 3.6. Our measurements show that the velocity amplitude is predicted within
40%. However, in general we will not observe standing waves. Therefore, the velocity amplitude can be calculated using the acoustical model described in section 3.4.1
using just the two pressure transducers in the turbine meter. However, these results
show similar accuracy as the standing wave approximation described above.
In general the acoustical signal should be distinguished from the pressure fluctuations induced by vortices. For plane waves this can be done by using more than two
microphones. Using a few microphones at one specific distance from the rotor one
can find the plane wave contribution by selecting the coherent part of the signal of
the 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 perturbations and with flow perturbations generated by the loudspeaker or the siren. The
effect on the rotation speed, averaged over one revolution, is determined from a visual examination of the plots of the signals as shown in figure 3.9. Using a ruler
deviations in the signal of 0.1% can be determined.
Epuls =
ω − ω0
,
ω0
(3.24)
66
3. Response of the turbine flow meter on pulsations with main flow
s ire n o ff
3
1 .9 1
2 .8
1 .8 9
1 .8 8
w - w
2 .4
v e lo c ity ( m /s )
v e lo c ity ( m /s )
2 .6
1 .8 6
w
1 .8 5
2
1 .8 3
0
2 0
4 0
tim e ( s )
6 0
lo u d s p e a k e r o n
1 .8 2
8 0
0
2 0
4 0
6 0
tim e ( s )
(a)
1 .7 8
1 .9 3
v e lo c ity ( m /s )
v e lo c ity ( m /s )
w - w
1 .7 5
0
1 .9 0 5
1 .9
0
2 0
w
1 .7 3
0
4 0
tim e ( s )
6 0
1 .7 1
8 0
0
2 0
4 0
6 0
tim e ( s )
8 0
1 0 0
1 2 0
(d)
1 .8 6 5
1 .9 4
1 .8 6
1 .8 5 5
1 .9 3 5
1 .9 3
w - w
1 .9 2 5
1 .9 2
w
0
w
1 .8 5
0
1 .8 4 5
v e lo c ity ( m /s )
v e lo c ity ( m /s )
0
1 .7 2
(c)
0
1 .8 4
lo u d s p e a k e r o ff
lo u d s p e a k e r o n
1 .8 3 5
1 .8 3
s ire n o ff
w - w
1 .8 2 5
1 .9 1 5
1 .9 1
0
lo u d s p e a k e r o n
1 .7 4
1 .9 1 5
w
1 2 0
lo u d s p e a k e r o ff
1 .7 6
1 .9 1
1 0 0
1 .7 7
1 .9 2 5
w - w
8 0
(b)
s ire n o ff
1 .9 3 5
1 .9 2
0
0
1 .8 4
w
0
w - w
1 .8 7
0
2 .2
1 .8
lo u d s p e a k e r o ff
1 .9
0
0
1 .8 2
2 0
4 0
tim e ( s )
(e)
6 0
8 0
1 .8 1 5
0
2 0
4 0
6 0
tim e ( s )
8 0
1 0 0
1 2 0
(f)
Figure 3.9: The velocity measured with the turbine meter. The black oscillating line corresponds to the instantaneous reading of the flow meter. The smooth white line
represents the measured velocity averaged over one rotation. The left figures
show typical measurements for perturbations generated by the siren. The siren
is turned on the first 30 seconds and then turned off. The right figures show typical results of measurements for the case in which perturbations are generated
with the loudspeaker, starting with the speaker turned off, then turned on, subsequently turned off. The figures show some examples of different pulsation levels
from extreme high (a,b) to low (e,f). Due to the long transient in the siren the
influence of low pulsation levels cannot be detected (e) while they are still very
clearly observable when the loudspeaker is used (f).
3.6. Measurements
67
where ω is the angular velocity of the rotor while measuring pulsating flow and ω0
is the angular velocity of the rotor for flow without pulsations. The variations of the
pressure in the reservoir, induces slow mass flow variations during a measurement.
This is the main cause of inaccuracies in determining the measurement error of the
flow meter.
The siren needs some time to reach a constant pulsation frequency, therefore these
measurements are started with the siren already turned on. After the siren is turned
off, this effect can also be seen. The slowing down of the siren causes the frequency
to decay, possibly inducing pulsations that can momentarily cause a large oscillations
in the measuring error during the transition. The influence of the pulsations is more
accurately determined by using the loudspeaker. The loudspeaker is turned on and
off during the measurement without complex transitional behaviour, making it easier
to measure the effect of the pulsations. However, the loudspeaker could only be used
at low flow velocities, up to 5 m/s in the main pipe. Measurements carried out with
the 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 turbine meter, measurements were carried out at resonance frequencies of the set up
between 24 Hz and 730 Hz and amplitudes of velocity pulsations ranging from
small, uac /u0 ≈ 0.01, to very high amplitudes, uac /u0 ≈ 2. The turbine flow meter
used in the set up (Instromet type SM-RI-X G250) has a flow range from 20 to 400
m3 /h (5.6 10−3 m3 /s to 0.11 m3 /s), this corresponds to a velocity in the pipe of
u0 = 0.7 m/s to 13.3 m/s. In our measurements velocities were varied from u0 =
0.5 m/s up to 15 m/s. In figure 3.10 and in figure 3.11 measurements are shown
for a pulsation frequency of f = 164 Hz for different mainstream velocities. Both
figures show exactly the same data set, however, in figure 3.10 the data is shown on
a 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 amplitude
extending over two decades and the range of main stream velocities, the measurements are still in fair agreement with the quasi-steady theory presented in 3.2.1. We
observe less than 40% deviation from the theory. By looking at the data, we can see
that the deviation from the quasi-steady theory increases for decreasing main flow
velocities. Data obtained for pulsation frequencies of 24, 69, 117, 360 and 730 Hz
are shown in Appendix C. In the section below the effect of the Strouhal number and
the Reynolds number will be investigated systematically.
68
3. Response of the turbine flow meter on pulsations with main flow
1
10
0
relative measurement error, E
puls
10
−1
10
quasi−steady theory
u0 = 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
−2
10
−3
10
−4
10
−5
10
−2
−1
10
0
1
10
10
relative acoustic amplitude, |u’|/u
10
0
Figure 3.10: The relative measurement error, Epuls , as a function of the relative amplitude
of the pulsations, |u′ |/u0 , for measurements at a frequency of 164 Hz. Plotted
using double logarithmic scale
0.5
relative measurement error, E
puls
0.45
0.4
0.35
quasi−steady theory
u0 = 15 m/s
u0 = 1 m/s
u0 = 1.3 m/s
u = 1.5 m/s
0
0.3
0.25
u0 = 1.7 m/s
u0 = 2 m/s
u = 3 m/s
0
0.2
0.15
u = 5 m/s
0
u0 = 7 m/s
u = 10 m/s
0
0.1
0.05
0
0
0.2
0.4
0.6
0.8
relative acoustic amplitude, |u’|/u
1
0
Figure 3.11: The relative measurement error, Epuls , as a function of the relative amplitude
of the pulsations, |u′ |/u0 , for measurements at a frequency of 164 Hz.
3.6. Measurements
3.6.1
69
Dependence on Strouhal number
In order to verify the range of validity of the quasi-steady theory, measurements have
been carried out for a wide range of Strouhal numbers, Sr = f Lublade
, where f is the
0
frequency of the pulsations, Lblade is the length of a rotor blade at the tip and u0 is
the main flow velocity at the position of the rotor. It is expected that for low Strouhal
numbers the quasi-steady theory is valid. From figure 3.11 (and Appendix C), it
is found that measurements for a given fixed frequency, f , and a fixed mainstream
velocity, u0 , have a quadratic dependence on the relative velocity amplitude, uac /u0 .
To investigate the dependence of the deviation in measured volume flow and actual
flow Epuls , a quadratic function:
Epuls = a
uac
u0
2
,
(3.25)
was therefore fitted through the measured data at a given frequency, f , and flow
velocity, u0 using least-square fitting. The parameter a will be referred to as the
”quadratic fit parameter”. This parameter is 21 for the quasi-steady theory. An example is shown in figure 3.12 for measurements at main stream velocity u0 = 1 m/s
and at frequency of pulsation f = 164 Hz.
We will consider only measurements with a relative amplitude uac /u0 < 1
to obtain the quadratic dependence, because higher amplitudes no longer show the
quadratic dependence. Measurements with relative amplitudes uac /u0 > 1 are discussed separately in section 3.6.3.
In figure 3.13 the quadratic fit parameter, a, for the measurement at pulsation
frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 m/s to 15
m/s are plotted against Strouhal number. The error bar gives the 95% confidence
level for the quadratic fit through the measured data. It is an indication for the quality
of the quadratic fit. In figure 3.13 the data measured using the siren are solid symbols.
The trends in the pulsation error measured with the siren and loudspeaker do not
differ from each other. However around SrLblade = O(1), the siren data seems to have
a slightly higher quadratic fit parameter than the loudspeaker data. An explanation
for this could be that most of the measurements using the loudspeaker are for smaller
relative amplitudes compared to the measurements using the siren. This indicates
that at low amplitudes the pulsation error, E, is probably not exactly quadratically
dependent on the velocity amplitude.
The figure shows a clear Strouhal dependence, where the deviation from the actual flow decreases with increasing Strouhal number. However, the deviation from the
actual volume flow stays within 40% of the quasi-steady theory for Strouhal number,
SrLblade , up to 2.5. Using regression, an equation is obtained to predict the depen-
70
3. Response of the turbine flow meter on pulsations with main flow
relative pulsation error, Epuls
0.5
0.4
quasi−steady theory
measured data
fitted equation, a(uac/u0)2
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
relative acoustic amplitude, u /u
ac
1
0
Figure 3.12: A quadratic fit, Epuls = a(uac /u0 )2 with a = 0.35 is shown for the measurement data at a mainstream velocity u0 = 1 m/s and with pulsations of
frequency f = 164 Hz. Quasi-steady theory gives a = 1/2.
dence of Epuls on the Strouhal number, SrLblade :
1
Epuls = −0.3672 SrL5 blade + 0.7407 , for 0.05 ≤ SrLblade ≤ 2.5 .
(3.26)
This is a purely empirical relationship between the deviation and the Strouhal number, which cannot be explained theoretically. It is interesting to note that for SrLblade <
0.2 we find a > 21 . 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 the
low frequency data. The measurements at a pulsation frequency of 360 Hz all have
a quadratic fit parameter, a, around the 0.5. These measurements are closer to the
deviation, Epuls , found by the quasi-steady theory than the equation found for the
lower frequencies. The measurements at a pulsation frequency of 730 Hz show
the same quadratic fit parameter for Strouhal numbers of around 6. However, at a
Strouhal number of about 10 the data seems to support the empirical relation found
using the lower pulsation frequencies. It is possible that these frequencies correspond
to mechanical resonant frequencies of the turbine meter causing a different behaviour
of the rotor.
3.6. Measurements
71
0.65
24 Hz
69 Hz
117 Hz
164 Hz
quadratic fit parameter, a
0.6
0.55
quasi−steady theory
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
0.5
1
1.5
Sr
2
2.5
L
blade
Figure 3.13: The quadratic fit parameter a is plotted against Strouhal number for pulsation
frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 to
15 m/s. The error bars represent the 95% confidence level for the quadratic
equation fitted through the measured data. The solid line shows the quadratic
fit parameter, a = 0.5, for the quasi-steady theory, the dashed line is a function
fitted through the data found from the present measurements. The solid symbols
represent the measurements using the siren, the open symbols the measurements
using the loudspeaker.
0.8
360 Hz
730 Hz
quadratic fit parameter, a
0.7
0.6
quasi−steady theory
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Sr
8
10
12
L
blade
Figure 3.14: The quadratic fit parameter a as a function of Strouhal number for pulsation
frequencies of 360 and 730 Hz. The ’♦’ show the data of measurements at
pulsation frequencies of 360 Hz, the ’’ show the data of measurements at
pulsation frequencies of 730 Hz. The solid symbols represent the measurements
using the siren, the open symbols the measurements using the loudspeaker.
72
3. Response of the turbine flow meter on pulsations with main flow
Besides this, as explained in section 3.4.3 for high frequencies small errors in the
pressure can cause large errors in estimated acoustical velocity. This could explain
some of the differences in Strouhal number dependence. Another possibility is that
the measurements at these frequencies are difficult is the presence of acoustic resonance. When the length of the constriction caused by the core of the meter matches
about half the wave length of the acoustical waves, there will be a resonance in this
pipe segment. The length of this constriction is about 25 cm, i.e. this would give
resonance frequencies of 680 Hz. For this resonance the rotor is close to a pressure antinode which corresponds to conditions in which the acoustical velocity at the
turbine is difficult to determine.
To exclude the possibility that the hollow space inside the flow straightener could
act as a Helmholtz resonator, this area was filled with foam. This did not change the
observed response of the flow meter to pulsations.
Care was taken to prevent these problems occurring at high frequencies by excluding measurements for which the rotor was close to a pressure antinode.1 To be
able to draw more conclusions for the deviation, Epuls , at Strouhal numbers greater
than 2.5, additional measurements are necessary.
At high frequencies, 367 Hz and 730 Hz, 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 2 m/s. These errors do not always reproduce. We suspect here a combination of mechanical vibration and friction. During
our tests dust particles were present in the flow and this affected the friction in the rotor. However, tests after cleaning the rotor indicated that this had only a minor effect
on most of our data. No significant effect is found for u0 > 2 m/s.
3.6.2
Dependence on Reynolds number
To investigate if there is also a dependence of the deviation, Epuls , on the Reynolds
number, ReLblade , the residual of the Strouhal number dependence predicted by the
empirical relation (equation 3.26) and the quadratic fit parameter found for the measurements is plotted as a function of Reynolds number, ReLblade for pulsation frequencies of 24, 69, 117 and 164 Hz (see figure 3.16).
Figure 3.16 shows no significant correlation, between the Reynolds number and
the difference between the measurements and the empirical relation for the Strouhal
number dependence. We conclude that that there is no significant dependence of the
Reynolds number, ReLblade , on the deviation, Epuls .
1
Note that all measurements between 360 Hz and 730 Hz have been rejected because of a very
large scatter in the quadratic fit coefficient a, which was related to difficulties in the measurement of the
acoustical velocity.
3.6. Measurements
73
0 .9 6 6
0 .9 6 4
lo u d s p e a k e r o ff
0 .9 6 2
v e lo c ity ( m /s )
0 .9 6
0 .9 5 8
0 .9 5 6
lo u d s p e a k e r o n
0 .9 5 4
0 .9 5 2
0 .9 5
0 .9 4 8
0 .9 4 6
0
2 0
4 0
6 0
tim e ( s )
8 0
1 0 0
1 2 0
Figure 3.15: Some measurements at low velocities and high frequency show a negative measurement error. In this plot the velocity of the flow measured by the turbine
meter is given, the smooth white line represents the velocity averaged over one
revolution of the turbine meter. The loudspeaker is turned on after 30 s inducing
pulsations of 730 Hz, after another 30 s the loudspeaker is turned off.
3.6.3
High relative acoustic amplitudes
Several measurements were carried out at relative pulsation amplitudes larger than
unity; 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, it
is 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 extrapolation of the quadratic dependence found for uac /u0 < 1. The difference with
this quadratic dependence is still small for relative acoustic amplitudes uac /u0 ≈ 1
and increases for increasing amplitude. Typical measurement data is shown in figure 3.17. At pulsation levels, uac /u0 ⋍ 2.5, the quasi-steady theory overestimates
the effect of pulsations by about a factor 2. While for low amplitudes the effect of
pulsations is overestimated by a factor 1.4.
74
3. Response of the turbine flow meter on pulsations with main flow
0.1
0.08
0.06
a−a
fit
0.04
0.02
0
−0.02
−0.04
−0.06
0
1
2
3
4
Re
L
blade
5
6
7
4
x 10
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 section
3.6.1, af it ,is plotted as a function of Reynolds number for pulsation frequencies
of 24, 69, 117 and 164 Hz.
3.6.4
Influence of the shape of the rotor blades
The influence of the shape of the blade was investigated by replacing the standard
rotor with a rotor a with different blade shape. The original rotor has blades with a
rounded upstream leading edge and a chamfered trailing edge. The rotor was replaced
by 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 pulsating
flow some of the measurements carried out with the standard rotor are repeated using
the new rotor. Figure 3.19 shows the results of the measurements carried out at
a pulsation frequency of 164 Hz and mainstream velocities of u0 = 1, 5 and 15
m/s, compared to the measurement data obtained for the standard rotor. Within
the accuracy of the measurement no difference was found. To verify this further a
quadratic fit as explained in section 3.6.1 was made and this parameter was plotted
against Strouhal number, SrLblade for low frequencies (f = 24, 69, 117 and 164 Hz)
(figure 3.20). Again, we see that within the accuracy level of the measurements there
is no difference between the deviation of the volume flow measurement for the rotor
with blades with rounded leading edges and the rotor with blades with chamfered
leading edges.
3.7. Conclusions
75
2
relative pulsation error, E
puls
1.8
1.6
quasi−steady theory
measurement data
fitted equation, a(u /u )2
ac
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
relative acoustic amplitude, u /u
ac
2.5
0
Figure 3.17: The deviation, Epuls as a function of the relative acoustic amplitude, uac /u0 . A
quadratic fit, Epuls = a(uac /u0 )2 derived for uac /u0 < 1 (with a = 0.35) is
shown for the measurement data at a mainstream velocity u0 = 1 m/s and with
pulsations of frequency f = 164 Hz.
3.7
Conclusions
The effect of the pulsating flow on a turbine flow meter has been investigated experimentally and results have been compared to the results of a simplified quasi-steady
model. A set up was built making it is possible to induce pulsations with a frequency
from 24 Hz to 730 Hz, relative acoustic velocity amplitudes, uac /u0 , from 2 × 10−2
up to 2 and volume flows ranging from the minimum to the maximum flow specified
by the manufacturer, i.e. from 20 to 400 m3 /h. Multi-microphone measurements
have been used to determine the amplitude of the velocity pulsation at the rotor. The
error caused by the pulsations is obtained from the comparison of the rotation speed
of the rotor in presence of pulsations with the one in the case that there are no pulsations. The measurements show that the simplified quasi-steady theory gives a fair
approximation of the error caused by the pulsations. The measurements agree with
the theory within 40% for nearly all measurements, even for measurements at high
relative acoustical amplitudes. We found that the error caused by pulsations is dependent on Strouhal number. For SrLblade < 2.5 an empirical relation was found for
the dependence of the error on the Strouhal number. Globally one expects that the
influence of pulsations should decrease with increasing pulsation Strouhal number.
76
3. Response of the turbine flow meter on pulsations with main flow
(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 a
chamfered leading edges used in the measurements
This corresponds to our observations. As of yet no physical explanation is found for
this 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 measurement error caused by the pulsations is not significantly dependent on the Reynolds
number. The shape of the upstream edge of the rotor blades does not influence the
Strouhal number dependence of the systematic error induced by the pulsations.
This study stresses the importance of determining the acoustical velocity at the
rotor for a correction of measurement errors due to pulsations. Measurements with
local velocity probes such as hot wires are difficult to use because they do not distinguish between vortical perturbations and acoustical waves. Acoustical waves can be
detected by means of microphones mounted flush in the wall. This would however
involve multiple microphones at a certain position to allow the detection of the plane
waves by cross-correlation method analogous to microphone array techniques.
3.7. Conclusions
77
0.18
relative measurement error, Epuls
0.16
0.14
0.12
quasi−steady theory
u = 1 m/s; round l.e.
0
u = 1 m/s; sharp l.e.
0
u0 = 5 m/s; round l.e.
u = 5 m/s; sharp l.e.
0
0.1
0.08
u = 15 m/s; round l.e.
0
u = 15 m/s; sharp l.e.
0
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
relative acoustic amplitude, |u’|/u0
0.6
Figure 3.19: The relative measurement error, Epuls , is plotted against the relative pulsation amplitude, |u′ |/u0 , for measurements with the ’new’ rotor with chamferedleading-edge blades and the standard rotor with rounded-leading-edge blades.
The data is for measurements at a pulsation frequency of 164 Hz and mainstream velocities, u0 = 1, 5 and 15 m/s. The solid symbols are the data measured with the rotor with chamfered-leading-edge blades. The plot shows that
within the measurement accuracy there is no difference in behaviour for the two
rotors.
0.65
quadratic fit parameter, a
0.6
0.55
quasi−steady theory
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
0.5
1
1.5
2
2.5
SrL
blade
Figure 3.20: The quadratic fit parameter a is plotted against Strouhal number for pulsation
frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 to
15 m/s. The ’o’ represents the measured data of the standard rotor with blades
with rounded leading edges and ’*’ represents the data measured with the ’new’
rotor with blades with chamfered leading edges. Again, no difference can be
found within measurement accuracies for the two rotors.
78
3. Response of the turbine flow meter on pulsations with main flow
4
Ghost counts caused by pulsations without
main flow
4.1
Introduction
Turbine flow meters are often placed in measurement manifolds, consisting of several
runs (side branches). At low volume flows some of the branches in these manifolds
are closed. Closed side branches can form resonators which are driven by vortex
shedding at the junction with the main pipe (Peters, 1993). Flow pulsations in these
manifolds can then affect turbine meters in open pipes, but can also induce ghost
counts. This corresponds to spurious flow measurement of the turbine meter in closed
side branches where there is no main flow. These ghost counts start above a critical
pulsation level.
The aim of this chapter is to obtain a better understanding of these spurious
counts. In the first part of this chapter1 a theoretical model is presented which
explains the occurrence of these spurious counts in the limit of infinitesimally thin
turbine blades. The predicted threshold for the occurrence of spurious counts is compared to experimental data at various gas pressures in the range from 1 to 8 bar.
In the second part of this chapter, a numerical and experimental study of the flow
around the edge of a turbine blade is presented. Aim of this study is to predict the
influence of the thickness and the shape of the turbine blade on the onset of spurious
counts. An experimental setup has been built to simulate the flow around a model of
a blade edge. The vortex shedding at the edge has been visualised. On the blade the
surface pressure has been measured. These measurements have been compared with
predictions of a discrete vortex model.
1
The first part of this chapter has been published with some minor changes in the Journal of Fluids
and 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
4.2
4.2.1
4. Ghost counts caused by pulsations without main flow
Onset of ghost counts
Theoretical modelling of ghost counts
The spurious count behaviour of a turbine meter can be explained by considering the
forces acting on an aerofoil in an oscillating flow. The blades of the turbine rotor
most commonly used in gas transport systems have a rounded leading edge and a
sharp trailing edge (see figure 4.1). This difference in edge shape causes the spurious
rotation.
le a d in g e d g e
u '
L
b la d e
t b la d e
tra ilin 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 trailing
edge of the blade. The flow separation at the sharp edge in an oscillating flow can
be seen in Schlieren visualisations (see figure 4.2). In the following we neglect the
interaction between the blades in the rotor.
Centrifugal forces in a potential flow around the edge of a plate cause a low
pressure at the surface of the blade edge, which results in a force directed along the
blade, a ”suction force”, which will be called the edge force. In case of a sharp edge
vortex shedding reduces this edge force, while for a rounded edge it remains present
as long as the flow remains attached. This leads to a net force on the blade. This
force brings about a torque on the rotor. Spurious counts start when this torque is
large enough to compensate the torque due to the static friction forces. This analysis
is restricted to the case of a harmonic acoustical oscillation with frequency f (Hz)
4.2. Onset of ghost counts
81
Figure 4.2: Schlieren visualisations of the flow separation at the sharp edge at a Strouhal
number, Srtblade = O(1)
and amplitude uac (ms−1 ) of the particle velocity: u′ = uac cos(2πf t). Furthermore,
the turbine is considered at the condition that it does not yet rotate, so that the blades
have a fixed position.
Important dimensionless numbers for this problem are the Helmholtz number,
He = f tblade /c , the Strouhal number, Srtblade = f tblade /uac , the Reynolds number,
Re = uac Lblade /ν and the ratio, tblade /Lblade , of the thickness, tblade , of the blade
compared with the length, Lblade . In these numbers ν (m2 s−1 ) is the kinematic
viscosity and c (ms−1 ) the propagation speed of sound waves.
It is assumed that the flow is attached and that the viscous boundary layers are thin
(Re >> 1). Hence, the flow around a blade of the turbine meter can be described
with potential flow theory corrected for the effect of boundary layers. The flow is
assumed to be locally incompressible, because the rotor is small compared to the
acoustical wave length and the amplitude, uac , is small compared to the speed of
sound (He << 1 and M = uac /c << 1). The Strouhal number, Srtblade gives an
indication of the blade thickness compared to the acoustical displacement of particles.
If the Strouhal number SrLblade = Srtblade Lblade /tblade is small, the blade length is
small compared to the acoustical particle displacement and the flow can be assumed
to be quasi-steady. If the Strouhal number is of order unity, Srtblade = O(1) with
tblade /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 shedding
is negligible except for very sharp edges. In that case the blade thickness is not the
relevant length scale. This is illustrated in figure 4.3.
In practice the ratio, tblade /Lblade , of the thickness of the blade compared to the
82
4. Ghost counts caused by pulsations without main flow
SrtbladeLblade/tblade<<1
Srtblade = O(1),
tblade/Lblade=O(10 -1 )
Srtblade >> 1,
tblade/Lblade=O(10 -1 )
u'
Figure 4.3: Influence of Strouhal number and blade length to thickness ratio L/tblade on the
flow. SrLblade ≪ 1 corresponds to quasi-steady flow. When SrLblade > 1, but
Srtblade < 1 we have strong vortex shedding at the edges. For SrLblade ≫ 1 and
Srtblade > 1 we have local vortex shedding at the sharp edges.
length, is small, tblade /Lblade = O(10−1 ). In this theory, the blade is modelled
as a flat plate. In a first model the flow separation at the sharp edge is modelled
by means of a single point vortex and by applying the Kutta condition (also called
Kutta-Joukowski condition) at the sharp edge. This corresponds to the model of
Brown and Michael (1954) for flow separation at the leading edge of a slender delta
wing. The force on the blade is found by integration of the pressure distribution on
the plate. The singular flow around the sharp leading edge results in a finite so-called
edge force which is the result of an infinite low pressure applied on a zero surface
(Milne-Thomson, 1952). A second model is considered for the limit case, for which
the Strouhal number is very large, Srtblade >> 1. Here a potential flow is considered
without flow separation, but the contribution of the sharp trailing edge to the net
hydrodynamic force on the blade is removed. The idea is that the vortex shedding
at this sharp edge has removed the local flow singularity without affecting the global
flow around the blade.
Theory for the case SrLblade = Srtblade (Lblade /tblade ) = O(1)
The theory for the case in which the Strouhal number is order unity is considered
first. In a point vortex model, the vortex sheet generated by flow separation at the
sharp edge is represented by a single point vortex of varying circulation. The vortex
is 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 the
sharp edge. The Kutta condition requires the velocity to be finite at the sharp edge.
In a real flow, this means that the flow leaves the edge tangentially, accounting for
viscous effects. In a point vortex model this implies that at the edge a stagnation point
is assumed. The point vortex moves with the flow. Application of the Kutta condition
4.2. Onset of ghost counts
83
implies then that the circulation changes with time. The convection velocity of the
point vortex is calculated by means of potential flow theory. For this calculation a
free vortex is assumed. This assumption is in contradiction with the time dependence
of the circulation of the vortex. This induces a spurious force that will be neglected
further (Rott, 1956). This error will appear not to be critical for our results.
ξ-plane
z-plane
Im[ ξ]
u'
α
α
Im[z]
Γv
Γv
A
Re[z]
Re[ξ]
Figure 4.4: Flat plate in the z-plane and transformed to a circle in the ξ-plane
The flow potential is calculated using a complex potential and conformal mapping. A circle with radius A, in the ξ-plane, is transformed in a flat plate of length
Lblade = 4A, in the z-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 written
as:
Φ = u′ ξe−iα +
iΓv
A2 ξv
u′ A2 iα iΓv
e −
ln(ξ − ξv ) +
ln(ξ −
).
ξ
2π
2π
|ξv |2
(4.2)
where α 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 potential for a circular cylinder in a parallel flow after applying the Milne-Thomson circle
theorem (Milne-Thomson, 1952). The third and the fourth term are the contributions
of the vortex and that of the mirror-imaged vortex at ξ = A2 /ξv∗ , also found using
2
Note that using a two-dimensional infinite cascade representation of the rotor we could use conformal mapping from a cascade to a circle proposed by Durant (1963). This would allow to take the
interaction between the rotor blades into account.
84
4. Ghost counts caused by pulsations without main flow
Milne-Thomson’s circle theorem. Here ξv is the position of the vortex in the transformed plane and Γv the circulation of the vortex. The circulation of the vortex is
calculated using the Kutta-condition at ξ = A:
dΦ =0.
dξ ξ=A
(4.3)
At the first step the vortex is shed, the position of this vortex is calculated using the self-similar solution, given by Howe (1975) for an impulsively starting flow
around a semi-infinite plate. The velocity of the vortex, uv , is calculated in the following steps using the following equation, assuming the vortex is a free vortex:
u∗v =
.
dΦ iΓv
dz iΓv d2 z . dz 2
dzv∗
dΦ
,
=
+
+
lim
dt
dz z=zv ξ→ξv dξ
2π(ξ − ξv )
dξ
2π dξ 2
dξ
where zv is the position of the vortex, ξv the position of the vortex in the transformed
plane, ∗ indicates the complex conjugate, Φ the complex potential and Γv the circulation 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 integration scheme (Hirsch, 1988). The new circulation, Γv , of the vortex corresponding
to this new position, is calculated using the Kutta condition (equation 4.3). In this
model the circulation of the vortex vanishes when the acoustic flow is zero. At the
next time step a new vortex is shed. As an example the path of a single point vortex
and its circulation is plotted in figure 4.5 for a typical case under conditions for which
spurious counts were measured.
0.05
1
ac
Γ/(U L)
y/L
0.5
0
0
−0.5
−0.05
0.45
0.5
x/L
0.55
−1
0
0.2
0.4
0.6
0.8
1
t/T
Figure 4.5: a) Calculated path of a single point vortex and b) calculated vortex strength for
SrLblade = 9
4.2. Onset of ghost counts
85
To calculate the torque on the turbine, the force on each blade is calculated numerically by integration of the pressure along the plate.
F =−
I
p · ndS = −
Z
without edge
pdS −
Z
pdS ,
(4.4)
edge
where p (P a) is the pressure and S (m2 ) the surface area. The pressure, p, in the
first term is obtained from the potential flow solution using the Bernoulli equation for
unsteady potential flow:
ρ
∂Φ 1 2
+ ρu + p = constant(t) .
∂t
2
(4.5)
The integration is carried out by means of the midpoint quadrature rule. For the
second term, around the singularity, a quasi steady-approximation is used (MilneThomson (1952) and Appendix D). As input for this theory a Taylor expansion of Φ
around the point ξ = −A is used.
Limit of the theory for SrLblade = Srtblade (Lblade /tblade ) >> 1
The limit of this theoretical model is considered next for SrLblade >> 1. In such a
case the effect of the vortex on the global flow around the blade is negligible except
for the flow near the sharp trailing edge. A great advantage of this simplified model is
that an explicit expression is obtained for the aerodynamic torque on the rotor without
the need to determine the details of the vortex path. As explained above the key of
this model is that it is assumed that there is only flow separation at the sharp trailing
edge of the flat plate. As a consequence, there is a finite velocity at this sharp edge
and therefore no edge force. At the leading, rounded edge there is no flow separation
and the velocity becomes infinite and this results in an edge force (figure 4.6).
a
F
e
a
F
e
Figure 4.6: Flow separation at the sharp edge for a flat plate in an oscillating flow
86
4. Ghost counts caused by pulsations without main flow
Because the velocity becomes infinitely large at the edge, convective flow acceleration is larger than the local time dependent flow acceleration. A quasi-steady
approximation can be used. The edge force is directed parallel to the plate and can be
calculated with potential flow theory (Milne-Thomson, 1952).3 The magnitude, Fe ,
of the edge force is:
Fe = −πρSblade u2ac sin2 α ,
(4.6)
where Fe (N is the edge force), Sblade (m2 ) is the surface area of the plate, uac
(ms−1 ) is the acoustic velocity amplitude and α is the angle between the blades and
the direction of the acoustical flow. The flow separation generates a vortex close to
the sharp edge. As explained above this vortex is assumed to have solely the effect
of removing the edge singularity in the flow field and as a consequence the edge
force. Flow separation also implies that the boundary layer vorticity is injected into
the main flow. This vorticity is assumed to be of small magnitude and confined to a
region close to the edge, therefore there will be no significant change of the global
circulation for the flow around the flat plate. If the flat plate is placed in a parallel harmonically oscillating flow, the flow will alternate between the left and right situation
in 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 taken
over one oscillating period will be zero. Consequently, the resultant averaged force
for the flat plate over one acoustic period can be simply calculated using the edge
force. With this edge force, it is possible to calculate the average torque, Tav , on the
blades.
Tav =
πρrav Sblade 2
uac sin3 α ,
8
(4.7)
where rav (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 for SrLblade = O(1) and the critical torque,
T2 , calculated with the model for the limit case SrLblade >> 1 is plotted against
the reciprocal Strouhal number, 1/SrLblade = uac /f Lblade , based on the length of
the 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 can be taken into account
by considering an infinite two-dimensional cascades of thin plates (Durant, 1963).
4.2. Onset of ghost counts
87
0 .4
0 .3 5
0 .3
0 .2
(T
2
-T
1
)/T
2
0 .2 5
0 .1 5
0 .1
0 .0 5
0
-0 .0 5
0
0 .0 5
0 .1
0 .1 5
1 /S r
0 .2
0 .2 5
0 .3
L
Figure 4.7: The difference (T1 − T2 )/T2 between the critical torque calculated with the two
models 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 resonant
response to compressor pulsations or by flow induced pulsations due to vortex shedding. In the experimental set up these flow oscillations are induced using a loudspeaker mounted within a closed pipe segment (figure 4.8).
Two set ups were used at Gasunie (Mulder, 2000), a small set up at atmospheric
pressure with a pipe diameter D = 100 mm, and a large set up, with a pipe diameter
D = 300 mm. In the large set up we had the possibility to vary the mean static pressure from 1 to 8 bar. In the small set up, the gas turbine meter (Instromet SM-RI-X
G250, see table 2.1) is placed between two PVC pipes with diameter D = 100 mm
and length Lp1 = Lp2 = 1.8 m. A loudspeaker (Visaton W100S) is placed at the
end of the pipe, while the other end is closed by a rigid plate. Four dynamical piezoelectric pressure transducers (Kistler type 7031) are placed at positions distributed
along the pipes. The pressure transducers are connected to charge amplifiers (Bruel
& Kjaer type 2635). Experiments with the small turbine meter were repeated at Eindhoven University of Technology (TU/e) with the set up described in section 3.3. In
the large set up the gas turbine meter (Instromet SM-RI-X G2500) is connected with
two pipes with diameter D = 300 mm and lengths of Lp1 = 6 m and Lp2 = 2 m.
The end of each pipe is sealed to be able to support a pressure up to 8 bar above atmospheric 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 the
loudspeaker can be changed, making it possible to modify the resonance frequency of
88
4. Ghost counts caused by pulsations without main flow
closed
side
branch
field conditions
main flow
laboratory experiment
D
loudspeaker
Lp1
Lp2
x=0
x
Figure 4.8: Field conditions compared with experimental set up
the system. Nine holes are made for placing dynamical pressure transducers (Kistler
type 7031) along the pipes, allowing an optimisation of the choice of the position of
the four available transducers.
To obtain a harmonic voltage signal, a signal generator (LMS Roadrunner) is
connected to a power amplifier (Bruel & Kjaer type 2706) from which the signal is
applied to the loudspeaker. The amplitude and the frequency can be adjusted separately. 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 measured using standard Elster-Instromet measuring equipment.
The pressure transducers are used to measure the acoustic pressure amplitude.
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 can
be assumed if the frequency, f , of the pulsations is much smaller than the cut-off
frequency, fc , for non planar waves (in the small set up fc = c0 /(2D) ≈ 1.7 × 103
Hz and for the large set up fc = c0 /(2D) ≈ 5.7 × 102 Hz). The pressure amplitude
data is used to calculate the acoustic velocity amplitude, uac , if a full reflection at the
closed end wall is assumed and thermal and friction losses in the pipe are neglected.
The acoustic velocity at a distance x from the end of the wall of the pipe is calculated
4.2. Onset of ghost counts
89
using the equation:
|u′ (x)| =
|p′ | sin kx
,
ρ0 c0 cos(kxm )
(4.8)
where |u′ | = uac (ms−1 ) is the acoustic velocity amplitude, |p′ | (P a) is the acoustic
pressure amplitude measured at a distance xm (m) from the end of the pipe wall, ρ0
(kgm−3 ) is the density of the propagation medium at ambient temperature, c0 ≈ 344
ms−1 is the acoustical wave propagation velocity in air at ambient temperature, k
(m−1 ) is the wave number and xm (m) is the distance between the end wall of the
pipe and the pressure transducer. To obtain the acoustic velocity, uac , at the blades of
the turbine flow meter, incompressibility is assumed and from the mass conservation
equation the following is obtained:
uac = |u(Lp2 )|
Sm
,
St
(4.9)
where uac (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 the
pipe where the measurement is taken and St (m2 ) is the cross sectional surface area
of the turbine flow meter. The flow through the turbine meter can be assumed incompressible, because the length of the turbine meter, Lt , is very small compared to the
wave length, λ (Lt ≪ λ). The frontal area of the blades of the turbine meter, which
is about 10% of the internal area of the turbine meter, is neglected in calculating St .
We used the values St = 5.7 × 10−3 m2 for the small meter (Instrometer G250) and
St = 4.9 × 10−2 m2 for the large meter (Instromet G2500). Measurements of uac
obtained from the four different pressure transducers agree with each other within
10%.
4.2.3
Experiments
Critical friction torque
Two different experiments have been carried out to obtain the critical static torque
above which rotation occurs: a dynamic experiment and a static experiment. The
equation of motion for the turbine flow meter is:
Irotor
dω
= Td + T f ,
dt
(4.10)
where Irotor (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 m2 s−2 ) is the driving
torque at the blades of the rotor and Tf (kg m2 s−2 ) the total friction torque. The
90
4. Ghost counts caused by pulsations without main flow
approximate driving torque has been derived in the previous section (see equation
4.7).
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 (the
friction in the oil and the friction of the shaft). The friction of the shaft can be divided
in static friction and dynamic friction.
To obtain the torque due to the dynamic friction, the rotor is accelerated to a
steady velocity by means of an acoustic field generated by a loudspeaker. When
the loudspeaker is turned off, the decay of the angular frequency is registered and a
fourth-order curve is fitted through the data. Subsequently, the torque is calculated
from 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 equation
4.6). The friction torque of the oil in the bearing has a linear dependency on the
rotational velocity . The friction of the shaft is assumed to be constant. In the limit of
vanishing rotational speed, the only contribution to the torque is the constant dynamic
friction of the rotor of the shaft. The value of this dynamic torque is found by taking
this limit to be approximately 6 × 10−6 N m for the small set up and 9 × 10−5 N m
for the large set up.
The static friction torque is measured using a small piece of adhesive tape fixed
at a known radius on the rotor. The tape induces a torque on the rotor, which can be
calculated from the weight and the position of the tape. The rotor is restrained and
released using a photograph shooter. The torque is increased step by step until the
release of the rotor induces rotation. With this method the critical static torque was
found to be 5.6 × 10−6 N m for the small set up and 1.0 × 10−4 N m for the large
apparatus. These values of the critical static torques agree within the experimental 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 counts
occur. This critical acoustic velocity is measured, by keeping the acoustical excitation
frequency, f , constant, while increasing the amplitude slowly. This can be done
either manually or by using a special function on the signal generator. When the
rotor starts to rotate the acoustic velocity is determined from the acoustic pressure
amplitude, p′ , as explained in the previous section. These results show considerable
deviations for different frequencies, but also for two consecutive measurements at the
same frequencies. The standard deviation is approximately 20% of the mean value.
Probable reasons for these deviations can be the varying static friction torque caused
by the unevenly distributed oil in the bearing and local roughness of the solid surfaces
4.2. Onset of ghost counts
91
in the bearing. Mechanical vibrations induced by the loudspeaker can also influence
the threshold for auto-gyration. It is also possible that they are caused by changes
that occur in the flow topology or by difficulties in the determination of the threshold
of rotation from the experimental data.
The mean critical acoustic velocities are determined for the small and the large
set up at their first and third resonance mode. For the small set up these frequencies
are f1 = 60 Hz and f3 = 210 Hz respectively, and for the large set up f1 = 30 Hz
and f3 = 100 Hz respectively. The critical acoustic velocity amplitude, uac , for the
large set up has been measured at four different static pressures, 1 bar, 2 bar, 4 bar
and 8 bar.
4.2.4
Comparing measurements with results of the theory
The results of the experiments can now be compared to the calculated data and to the
value of the critical pressure amplitudes in field conditions.
Strouhal number (Sr tblade=t blade f/uac )
1.2
1
0.8
0.6
Gasunie, large app. 30 Hz
0.4
Gasunie, large app. 100 Hz
Gasunie, small app. 60 Hz
Gasunie, small app. 210 Hz
0.2
TU/e, small app. 69 Hz
TU/e, small app. 165 Hz
0
0
1000
2000
3000
4000
5000
6000
7000
Reynolds number (ReLblade =uac Lblade/ν)
Figure 4.9: The critical Strouhal number plotted against the Reynolds number. The data
obtained at Gasunie (grey and solid circles and squares) complemented with the
data obtained at TU/e (open circles and squares) as described in section 4.3).
Figure 4.9 shows the critical Strouhal number, Srtblade = tblade f /uac , based on
the blade thickness, tblade , at which the rotation starts, plotted against the Reynolds
number based on the blade length, Re = uac tblade /ν, with ν the kinetic viscosity.
The critical Strouhal number, Srtblade , is of order unity, which corresponds to our
qualitative model of the effect of the blade thickness. When the vortex remains at
92
4. Ghost counts caused by pulsations without main flow
distances from the edge smaller than the blade thickness, it is not expected that this
affects the flow. Hence there is no rotation for Srtblade >> 1. A significant dependency of Srtblade on the Reynolds number can be observed, which we cannot explain
with the used model. We did actually not expect such a dependency.
Gasunie, large app. 30 Hz
1.8
ratio of measured and predicted
critical torque
Gasunie, large app. 100 Hz
1.6
Gasunie, small app. 60 Hz
Gasunie, small app. 210 Hz
1.4
TU/e, small app. 69 Hz
1.2
TU/e, small app. 165 Hz
1
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
7000
Reynolds number (ReLblade = uac Lblade/ν)
Figure 4.10: Ratio between the measured critical torque and the calculated critical torque
plotted against the Reynolds number. The data measured at Gasunie (grey and
solid circles and squares) complemented with the data measured at TU/e (open
circles and squares) which are described in section 4.3.
In figure 4.10 the ratio of the measured critical static torque and the predicted
torque calculated using the limit case model (SrLblade >> 1) for both set ups is
plotted against the Reynolds number, with the length of the blade as the characteristic
length. For the small set up the calculated critical torque is two times the measured
critical torque for the first resonance mode and five times the measured critical torque
for the third mode. For the large set up the calculated critical torque is 20% lower
than measured for the first mode and 1.4-1.7 times larger for the third mode. It would
be interesting to investigate whether these difference in behaviour for the first and
third acoustical mode of the pipe are related to difference in mechanical vibration
level. Such vibrations induced by the loudspeaker could affect the critical torque.
Using the results the critical torque can be estimated for field conditions of natural
gas transportation with a static pressure, p = 60 bar = 6 × 106 P a and a density of
the natural gas, ρg = 14 kg m−3 . The geometries of the pipe and of the turbine flow
meter are assumed to be similar to that of the large set up. Using equation 4.7 the
acoustical velocity, uac , is calculated, at which the critical static torque is reached and
spurious counts occur. This is found to be uac ≈ 7 × 10−2 ms−1 . This corresponds
4.3. Influence of vibrations and rotor asymmetry
93
to acoustical pressure amplitudes of the order of p′ = ρ0 c0 uac ≈ 103 P a with for
natural gas c0 ≈ 390 ms−1 . In field conditions measurements show spurious counts
with acoustic pressure amplitudes of 4 × 103 P a (Riezebos et al., 2001). This shows
that the model can provide a fair indication for conditions at which spurious counts
occur.
4.3
4.3.1
Influence of vibrations and rotor asymmetry
Vibration and friction
To verify the measurements described in section 4.2.3 the same measurements were
carried out in the set up built at Eindhoven University of Technology to measure the
effect on the turbine meter of pulsation with main flow (section 3.3). By closing the
valve and using the loudspeaker to induce pulsation, comparable measurements of
the critical acoustical velocity amplitude, uac , can be preformed. The turbine meter
in this set up is of the same type (Instromet SM-RI-X G250) as used in the small setup
described in section 4.2.2. In this set up, however, the loudspeaker is mechanically
disconnected from the pipe so that mechanical vibrations are reduced.
The amplitude of the critical acoustic velocity is determined at different frequencies by varying the amplitude step by step and waiting a few moments to observe
whether or not the rotor starts rotating. Around the critical amplitude the situation
can occur that the rotor begins to move but stops before a full revolution is completed.
This shows the dependence of the static friction torque on the rotor position. The dynamic friction torque was measured using the dynamical method described in section
4.2.3 and was found to be 1.5 × 10−5 N m.
The results are plotted in figures 4.9 and 4.10, and show a good agreement with
the other measurements. It should be noted that is found from this test and a check by
the manufacturer that the there is an increase of friction caused by the deterioration of
the 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 need
for higher pulsation to compensate the friction torque, the maximum pulsation is
induced corresponding with a Strouhal number of Srtblade = 0.048, using equation
4.7 of the limit model, this is equal to a torque of about Td = 2.8 × 10−4 N m.
4.3.2
Rotor blades with chamfered leading edge
In order to verify that the difference in shape of the edges of the blades of the rotor
causes the ghost counts, the rotor in the turbine meter was replaced with a rotor with
a blade profile with a chamfered leading edge (figure 4.11).
Again pulsations were induced to investigate the occurrence of ghost counts. As
94
4. Ghost counts caused by pulsations without main flow
(a) rounded leading edge
(b) chamfered leading edge
Figure 4.11: A schematic drawing of a rotor (a) with rounded leading edges and (b) with
chamfered leading edges used in the measurements
expected, no rotation of the rotor occurred. Also imposing an initial rotation of the
rotor is not sufficient to induce ghost counts. An initial rotation was induced by a
main flow, but when the main flow was stopped the rotor rotation decayed to zero
independently 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 the
influence 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 examined
more 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 equation
for two-dimensional flow neglecting viscous effects. For a more detailed description of this method we refer to Hofmans (1998) and Peters (1993). The ”vortexblob” method requires the specifications of separation points. We can use this in
order to identify the contribution of various vortices to the edge force. The vorticity field is modelled by vortex blobs, desingularised point vortices, and by applying
the Kutta condition at the separation points new vortex blobs are introduced. The
desingularisation of the vortex is used to avoid numerics-induced chaotic behaviour
in 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 panel
method. The dipole distribution on the walls is determined such that the normal flow
velocity 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 to
study the local flow around the edge of a single blade of blade thickness tblade = 1.00
cm. This was done by building a wooden box of 48.2 cm x 24.8 cm x 22.3 cm with a
wall thickness of 2.5 cm divided in the middle to create two separate compartments.
These two compartments are connected by means of a duct creating two connected
Helmholtz resonators (see figure 4.12). The volume V1 of the first compartment is
V1 = 8.15×10−3 m3 . The volume V2 of the second compartment is V2 = 8.35×10−3
m3 . This implies a small asymmetry in the volumes of the compartments in our set
up. The duct cross-sections are S = d x w with d = 4.5 cm and w = 10.0 cm.
The edge of the dividing wall is a model for the edge of the turbine blade. The tip
of the edge of is placed at a distance of 5.0 cm from the top wall closing the set
up. It is possible to change the shape of the blade edge. The first model has a sharp
chamfered edge with a bevel angle of 45◦ . The second model has a rounded edge
and the last model has a square edge. The experiments for the rounded edge show no
vortex shedding at the edge. We present here only the results for the sharp edge. The
internal 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 is
amplified be using a charge amplifier (Kistler type 5011). A third miniature pressure
transducer (Kulite type XCS-093-140mBARD) is placed in the wall right above the
blade 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 technique. The pressure can be measured at three positions by means of three miniature
pressure transducers (Kulite type XCS-093-140mBARD). The pressure transducers
are placed in the edge 5 mm from each other along the width of the edge model. The
slanted side of the edge model is a 0.4 mm thick plate covering the pressure transducers. Pressure holes of diameter 0.4 mm are made in this plate in order to measure the
pressure. The first hole is placed 2.0 mm from the tip of the edge model, the second
one 4.0 mm from the the tip and the third one 7.5 mm from the tip in the middle of
the slanted side. This is shown in figure 4.13. The pressures measured at the blade
edge, will be compared with the pressures obtained in the numerical simulations.
In the Schlieren visualisation refractive index contrast is obtained by heating up
the edge with an infrared lamp (Philips Infraphil HP 3608) similar to the technique
used by Disselhorst (1978). Before each flow visualisation the top plate of the set
96
4. Ghost counts caused by pulsations without main flow
p re s s u re tra n s d u c e rs
b la d e e d g e
p re s s u re tra n s d u c e r
w =
9 .9 5 c m
lo u d s p e a k e r
1 9 .8 c m
(a)
to p w a ll
p re s s u re tra n s d u c e rs
h o t w ire
e d g e p re s s u re
tra n s d u c e rs
p re s s u re tra n s d u c e rs
5 c m
p re s s u re tra n s d u c e r
S
lo u d s p e a k e r 1
V
2 0 .9 c m
u '1
u '2
d =
4 .5 c m
d =
4 .5 c m
V
1
t b la d e =
1 c m
1 7 .4 c m
lo u d s p e a k e r 2
1 9 .8 c m
2
2 1 .3 c m
(b)
Figure 4.12: Drawing (a) shows a 3d picture of the set up and (b) a cross-section of the set
up
4.4. Flow around the edge of a blade
97
0 .4 m m
2
2
p
3 .5
e d g e ,1
p
5
p
0 .4 m m
e d g e ,2
e d g e ,3
5
t
(a)
3 .0 m m
b la d e
= 1 0 .0
(b)
Figure 4.13: Drawing (a) shows a close-up of the pressure transducers in the edge shows
(dimensions in mm) 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 the
edge 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 unit generating
a TTL-pulse. The flash duration is typically 20 µs. This signal is used to determine
the frequency of the input signal and delayed to obtain a stroboscopic effect. Pictures
were taken by a digital high speed camera, the Philips INCA 311, with accurate
external triggering.
To calculate the velocity in the duct, u′ , we assume a uniform pressure in volume
V1 and V2 , so that:
Vi
dρ′
= −ρ0 Su′ − φloudspeaker ,
dt
i = 1, 2
(4.11)
with Vi the volume in one of the two sections of the box, S = d × w the surface
area of the connecting duct (see figure 4.12), ρ′ the density fluctuations, t the time,
ρ0 = 1.2kg m−3 the ambient density and φloudspeaker is the mass displacement of
the loudspeakers. Measurements have been carried out close to the resonance of the
set up at frequency f = 120 Hz. The quality factor determined by means of white
noise excitation of the setup is q = 3. We neglect φloudspeaker . Using ρ′ = p′ /c20 ,
with c0 = 344 m/s the speed of sound and p′ the pressure fluctuations, and assuming
harmonic flow, we find:
u′ = −
2iπf Vi p′
.
ρ0 c20 S
(4.12)
The velocity fluctuations, u′1 , have also been measured using a hot wire anemometer (Dantec type 55P11 wire diameter 5 µm with 55H20 support) placed 11.9 cm
from the top wall in the middle of the left duct. The hot wire measurements of the
98
4. Ghost counts caused by pulsations without main flow
velocity fluctuations agree within 25% with pressure fluctuation measurements when
using equation 4.12. During other measurements the hot wire was removed. The
amplitude of the velocity fluctuations, uac = |u′ |, are made dimensionless by means
of the Strouhal number, Srtblade = f tblade /uac , with f the frequency and tblade the
thickness of the blade. The time scale is related to the pressure in the reservoir V1 ,
assuming it has a cos (2πf t) time dependence for sin (2πf t) pressure fluctuations.
4.4.3
Measurements
Measurement have been obtained for acoustic flows with Srtblade = 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. At t/T = 0 the flow starts flowing from
V1 to V2 (left to right) and at t/T = 0.5 the flow changes direction and starts flowing
from V2 to V1 . The pictures, taken at t/T = 0.3 and 0.8, give an impression of the
size of the vortices for Srtblade = 0.2, 0.8, 3.2. For increasing Strouhal number the
size of the vortices decreases.
From the Schlieren visualizations, it is also observed that depending on the initial
conditions the flow can display two different modes (figure 4.15). In the first mode
the 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 flow
changes direction at t/T = 0.5. Both vortices move as a vortex pair away from the
edge (figure 4.15(a)). We will refer to this mode as ”mode 1”. In the second mode
the first vortex is created starting at t/T = 0.5 next to the edge model. A second
vortex of opposite sign is created starting at t/T = 0 above the slated side of the
edge. They move away from the edge as a vortex pair over the slanted side (figure
4.15(b)). This mode will be referred to as ”mode 2”. While it is possible to force
the flow in mode 2 (figure 4.15(b)), in most experiments mode 1 (figure 4.15(a)) is
dominant. At Srtblade = 1.6 the vortices that are created are small and it is no longer
possible to sustain mode 2 behaviour during a measurement. At even higher Strouhal
number, Srtblade = 3.2, the mode of vortex shedding changed spontaneously during
the measurement from one mode to the other and back. These measurements are not
included in this discussion.
The pressure is measured with the three pressure transducers at the edge, where
p′edge,1 , p′edge,2 and p′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), and p′top
the top wall pressure transducer right above the edge. The signal used to drive the
loudspeaker is used to determine the period of oscillation. The pressure signals are
phase averaged over 6 × 103 periods. The absolute mean pressure that is established
in the set up depends on leaks and is not reproducible. Because of the uncertainties in the mean pressure within the setup, the mean pressure measured at the top
4.4. Flow around the edge of a blade
99
(a) Srtblade =3.2
t/T=0.3
(b) Srtblade =3.2
t/T=0.5
(c) Srtblade =3.2
t/T=0.8
(d) Srtblade =3.2
t/T=1.0
(e) Srtblade =0.8
t/T=0.3
(f) Srtblade =0.8
t/T=0.5
(g) Srtblade =0.8
t/T=0.8
(h) Srtblade =0.8
t/T=1.0
(i) Srtblade =0.2
t/T=0.3
(j) Srtblade =0.2
t/T=0.5
(k) Srtblade =0.2
t/T=0.8
(l) Srtblade =0.2
t/T=1.0
Figure 4.14: Schlieren visualization of the flow for Srtblade = 3.2, 0.8 and 0.2 at t/T = 0.3,
0.5, 0.8 and 1.0.
100
4. Ghost counts caused by pulsations without main flow
(a) ”mode 1”
(b) ”mode 2”
Figure 4.15: Schlieren visualization of the two modes of vortex shedding: a) first vortex is
formed on the right side of the edge, the vortex pair moves to the left away from
the edge. b) first vortex is formed on the left, the vortex pair moves to the right.
has been subtracted from the signal. Figure 4.16 shows the dimensionless pressure,
p′ /(ρ0 c0 uac ), for mode 1 at Strouhal number Srtblade = 0.2, 0.4, 0.8 and 1.6. As a
reminder small pictures in the plot illustrates the position of the vortices.
The pressure fluctuations at the top wall show that there is no perfect standing
wave in the set up, because the pressure is not exactly in phase with p1 . The same
measurements are found by turning the top plate around 180◦ . Hence this is not due
4.4. Flow around the edge of a blade
101
0 .0 2
0 .0 4
0
0 .0 2
0
a c
/(r c
0
u
-0 .0 4
d g e ,2
-0 .0 6
p 'e
p 'e
d g e ,1
/(r c
0
u
a c
)
)
-0 .0 2
-0 .0 8
S r=
S r=
S r=
S r=
-0 .1
-0 .1 2
0
0 .2
0 .4
t/T
0 .6
0 .2
0 .4
0 .8
1 .6
-0 .0 2
-0 .0 4
-0 .0 6
0 .8
-0 .0 8
-0 .1
1
0
0 .2
(a) transducer 1
S r=
S r=
S r=
S r=
0 .0 4
0 .6
0 .8
S r=
S r=
S r=
S r=
0 .2
0 .4
0 .8
1 .6
1
0 .0 1
)
0 .0 2
0 .0 1 5
0 .2
0 .4
0 .8
1 .6
0 .0 0 5
0
0
p 'to p / ( r c
p 'e
d g e ,3
/(r c
0
u
u
a c
)
t/T
0 .2
0 .4
0 .8
1 .6
(b) transducer 2
0 .0 6
a c
0 .4
S r=
S r=
S r=
S r=
-0 .0 2
-0 .0 4
-0 .0 6
0
-0 .0 0 5
0
-0 .0 1
0 .2
0 .4
t/T
0 .6
(c) transducer 3
0 .8
1
-0 .0 1 5
0
0 .2
0 .4
t/T
0 .6
0 .8
1
(d) top wall transducer
Figure 4.16: Dimensionless pressure fluctuations measured for four different Strouhal numbers (Srtblade = 0.2, 0.4, 0.8 and 1.6) at the pressure transducers in the edge
model (a,b,c) for mode 1 vortex shedding and at the top of the set up (d).
102
4. Ghost counts caused by pulsations without main flow
to a misalignement of the top wall pressure transducer. This is not surprising, because
of the asymmetry of the flow caused by the edge model, the absorbtion of sound by
the vortices and because of the leakage in the set up.
For 0 < 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. In order
unsteady potential flow the equation of Bernoulli reads:
ρ
∂Φ 1
+ ρ|~v |2 + p = const(t)
∂t
2
(4.13)
R
were the flow potential is given by Φ = ~v · d~x. At high Strouhal numbers, such
as Srtblade = 1.6, the quadratic term 1/2ρ|~v |2 is almost negligible and we observe
an 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 a
decrease of the local pressure, when there is only local flow separation. For 0.7 <
t/T < 1 we see from the flow visualization that there is a strong flow separation
at 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 edge
pressure is lower than the pressure at the top wall indicating an edge force.
Figure 4.17 show the dimensionless pressure, p′ /(ρ0 c0 uac ) measured at these
two transducers, for mode 2 behaviours at Srtblade = 0.2, 0.4 and 0.8. As it was not
possible to sustain mode 2 vortex shedding behaviour for one measurement, measurements at Srtblade = 1.6 are not included. We observe that on average the edge
pressure 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 geometry
of the duct of the set up is modelled. The geometry of the edge and the duct around
the edge is discretised using 2800 panels. Around the tip of the edge the density of
the panels is increased. The calculations are carried out using 1000 equal time steps
per period. The desingularity parameter in the expressions for the velocity induced
by a vortex blob was chosen 10 times the time step.
Calculations with the blob method for the configuration considered fail to converge 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 very
crude approach. From the flow visualization it is observed that vortex shedding starts
close to t/T = 0. At that time the acoustical velocity is reversing from a flow from
V1 to volume V2 towards a flow from V2 to V1 . After the reverse of the acoustical
velocity at t/T = 0.5, a second vortex is shed containing opposite vorticity. The first
and the second vortex travel away as a vortex pair and seem to have little influence
4.4. Flow around the edge of a blade
0 .0 2
103
0 .0 3
S r= 0 .2
S r= 0 .4
S r= 0 .8
0 .0 1
0 .0 1
a c
/(r c
0
u
-0 .0 1
d g e ,2
-0 .0 2
p 'e
d g e ,1
/(r c
0
u
a c
)
)
0
p 'e
S r= 0 .2
S r= 0 .4
S r= 0 .8
0 .0 2
-0 .0 3
0
-0 .0 1
-0 .0 2
-0 .0 3
-0 .0 4
-0 .0 4
-0 .0 5
-0 .0 5
0
0 .2
0 .4
t/T
0 .6
0 .8
-0 .0 6
1
0
0 .2
(a) transducer 1
0 .6
0 .8
0 .0 1 5
S r= 0 .2
S r= 0 .4
S r= 0 .8
0 .0 4
1
S r= 0 .2
S r= 0 .4
S r= 0 .8
0 .0 1
0 .0 0 5
)
0 .0 2
0
0
p 'to p / ( r c
p 'e
d g e ,3
/(r c
0
u
u
a c
)
t/T
(b) transducer 2
0 .0 6
a c
0 .4
-0 .0 2
-0 .0 4
-0 .0 6
0
-0 .0 0 5
0
-0 .0 1
0 .2
0 .4
t/T
0 .6
(c) transducer 3
0 .8
1
-0 .0 1 5
0
0 .2
0 .4
t/T
0 .6
0 .8
1
(d) top wall transducer
Figure 4.17: Dimensionless pressure fluctuations measured for three different Strouhal numbers (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).
104
4. Ghost counts caused by pulsations without main flow
on the next vortex shedding. This allows to use the ”vortex-blob” method, during one
single 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 using
the vortex blob method, we calculated the difference between the pressure at the
three locations on the slanted side of the edge model and the pressure at the top wall.
The results are found in figure 4.19 for all three pressure transducers in the edge as
function of time. The left side shows the calculations using the blob method and on
the right side the measured pressures are shown.
The measurements and the numerical simulation show similarities in shape, however the fluctuations in the pressure obtained for the vortex blob method are larger
and the peaks are less wide.
A comparison is also done for mode 2 vortex shedding. For these calculations
t/T = 0 is defined as the start of the flow going from V2 to V1 , the flow starts
with the single vortex left of the edge. The difference between the pressures on the
edge at the three locations and the top pressure right above the edge is plotted for
Srtblade = 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 in
mode 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 remains close to
the edge. However, the visualisations do not show this vortex left behind. The same
effect takes place for mode 1 behaviour (figure 4.18(c,d,g,h and k)). This vortex has
less effect on the edge pressure, because it is not close to the edge as in the case of
mode 2.
4.5
Conclusions
Representing the rotor blade by a flat plate and the flow separation at the sharp edge
of the blade by a point vortex, a model is obtained allowing to predict the driving
torque on a rotor placed in an oscillatory flow. A simplified model is proposed for
high Strouhal number (SrLblade >> 1) which provides an explicit algebraic expression without the need to determine the details of the flow. Comparison between the
two models indicates that they are equivalent within the accuracy of the performed
experiments. The results show that the thickness of the plate is an important factor for occurrence of spurious counts. The presence of a thick trailing edge in turbine blades increases the critical acoustical pulsation amplitude above which spurious
counts appear. The model provides a prediction of the order of magnitude of the critical torque, and can be used to determine typical conditions for the occurrence of
4.5. Conclusions
105
3
3
t/T =
2.5
0.3
t/T =
2.5
0.5
3
3
t/T =
2.5
2
2
2
1.5
0.8
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0.5
0
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
−1
−1
−1.5
−1.5
−1.5
−1.5
−2
−2
−2
−1
0
1
2
−2.5
(a) Srtblade =3.2
t/T=0.3
3
−1
0
1
2
0.3
−1
0
1
−2.5
2
(c) Srtblade =3.2
t/T=0.8
3
3
t/T =
2.5
−2
−2.5
(b) Srtblade =3.2
t/T=0.5
t/T =
2.5
0.5
0.8
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
−1
−1
−1.5
−1.5
−1.5
−1.5
−2
−2
−2
1
2
−2.5
(e) Srtblade =0.8
t/T=0.3
3
−1
0
1
2
(f) Srtblade =0.8
t/T=0.5
0.3
t/T =
2.5
0.5
t/T =
0
1
−2.5
2
(g) Srtblade =0.8
t/T=0.8
−1
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
−1
−1
−1.5
−1.5
−1.5
−1.5
−2
−2
−2
2
(i) Srtblade =0.2
t/T=0.3
−2.5
−1
0
1
2
(j) Srtblade =0.2
t/T=0.5
−2.5
2
t/T =
2.5
2
1
1
0.8
2
0
0
3
t/T =
2.5
2
−1
1
(h) Srtblade =0.8
t/T=1.0
2
−2.5
2
−2
−1
3
3
t/T =
2.5
−2.5
1
2.5
2
0
0
3
t/T =
2.5
2
−1
−1
(d) Srtblade =3.2
t/T=1.0
2
−2.5
1
2
1
−2.5
t/T =
2.5
1
−2
−1
0
1
2
(k) Srtblade =0.2
t/T=0.8
−2.5
−1
0
1
2
(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 and 0.2
at t/T = 0.3, 0.5, 0.8 and 1.0.
106
4. Ghost counts caused by pulsations without main flow
0.02
−0.02
−0.08
edge,3
−p
−0.06
−0.08
−0.1
(p
−0.1
(p
edge,3
−0.04
top
0
−0.04
top
0
ac
−0.06
)/(ρ c u )
0
−0.02
−p
0
ac
)/(ρ c u )
0.02
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
−0.12
−0.14
−0.16
0
0.2
0.4
0.6
0.8
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
−0.12
−0.14
−0.16
0
1
0.2
0.4
t/T
(a) blob method, pedge,1
0.04
0.02
0.02
−0.06
0
ac
)/(ρ c u )
−p
edge,2
top
−0.04
−0.08
−0.1
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
−0.12
−0.14
0
0.2
0.4
0.6
0.8
−0.04
−0.08
−0.1
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
−0.12
−0.14
1
0
0.2
0.4
t/T
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
ac
)/(ρ c u )
0.03
0.02
0
0
ac
1
0.05
Sr=0.2
Sr=0.4
Sr=0.8
Sr=1.6
0.04
)/(ρ c u )
0.8
(d) measurement, pedge,2
0.05
top
−p
0
edge,1
top
0
edge,1
0.01
−0.01
−0.02
(p
−p
0.6
t/T
(c) blob method, pedge,2
(p
1
0
−0.02
(p
(p
edge,2
top
0
ac
)/(ρ c u )
−p
0
−0.06
0.8
(b) measurement, pedge,1
0.04
−0.02
0.6
t/T
−0.03
−0.04
−0.05
0
0.2
0.4
0.6
0.8
t/T
(e) blob method, pedge,3
1
−0.05
0
0.2
0.4
0.6
0.8
1
t/T
(f) measurement, pedge,3
Figure 4.19: The dimensionless difference between pressure pedge,1 , pedge,2 and pedge,3 and
the top wall right above the edge for Srtblade = 0.2, 0.4, 0.8 and 1.6. The
mode of vortex shedding is mode 2. On the left side the pressure difference as
predicted using the vortex blob method. On the right side the measured pressure
difference is shown.
4.5. Conclusions
0.01
0
0
ac
0
−0.02
−p
edge,1
−0.04
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
−0.06
0
−0.03
(p
top
−0.02
top
edge,3
−0.01
0
−0.04
)/(ρ c u
)/(ρ c u )
−p
−0.03
(p
)
0.01
−0.01
ac
107
0.2
0.4
0.6
0.8
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
−0.06
0
1
0.2
0.4
0.02
0.01
0.01
0
ac
)/(ρ c u )
top
−p
−0.03
(p
(p
−0.02
−0.04
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
−0.06
0
0.2
0.4
0.6
0.8
−0.04
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
−0.06
0
1
0.2
0.4
0.8
1
(d) measurement, pedge,2
0.04
0.03
0.03
0.02
0.02
ac
)/(ρ c u )
0.04
0.01
0.01
0
0
−0.02
(p
−0.03
−0.04
0.2
0.4
0.6
0.8
t/T
(e) blob method, pedge,3
−0.03
−0.04
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
−0.06
0
top
−0.02
0
−0.01
−p
top
−0.01
edge,1
0
edge,3
ac
)/( ρ c u )
(c) blob method, pedge,2
−p
0.6
t/T
t/T
(p
1
0
−0.01
edge,2
0
−0.03
−p
top
0
−0.02
0.8
(b) measurement, pedge,1
0.02
edge,2
ac
)/(ρ c u )
(a) blob method, pedge,1
−0.01
0.6
t/T
t/T
Sr=0.2
Sr=0.4
Sr=0.8
−0.05
1
−0.06
0
0.2
0.4
0.6
0.8
1
t/T
(f) measurement, pedge,3
Figure 4.20: The dimensionless difference between the pressure pedge,1 , pedge,2 and pedge,3
and the top wall right above the edge for Srtblade 0.2, 0.4 and 0.8. The mode of
vortex shedding is mode 2. On the left side the pressure difference as predicted
using the vortex blob method. On the right side the measured pressure difference
is shown.
108
4. Ghost counts caused by pulsations without main flow
spurious counts in field conditions. In view of its simplicity, the theory for the limit
case, SrLblade >> 1, is a useful engineering tool in the prediction of the occurrence
of spurious counts.
A model describing details of the flow around the sharp trailing edge of a rotor
blade is proposed. This model is based on the vortex blob method. In this model we
assume 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 after
one oscillation period (t > T ). The predicted vortex structure agrees qualitatively
with the flow visualization and the model predicts reasonably well the dependence
of the edge pressure on the Strouhal number. However, to obtain a prediction of the
edge force a more accurate numerical model is necessary.
5
Conclusions
5.1
Introduction
When used at ideal conditions, gas turbine flow meters allow measurement of the
volume flow with an accuracy of about 0.2%. During use systematic changes in
response can be induced by wear or damage of the rotor. Systematic errors can also
be induced by perturbations in the flow (non-uniformities, swirl or pulsations). This
study is limited to the behaviour of turbine flow meters for three different types of
flow:
• 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 acoustical
pulsations induce a fluctuation in velocity which is uniform across the rotor. In all
these cases the behaviour of the turbine meter is analysed by comparison of the results of analytical models with the results of experiments. Some of these simplified
models can be used to apply corrections or as support for design rules in engineering
applications.
5.2
Stationary flow
An ideal helicoidal turbine meter without mechanical friction nor fluid drag will have
a rotational speed, ω, proportional to the volume flow, Q: ω = KQ. In practice
the meter constant K 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 constant K on one side but also to achieve a rotor which is robust. This means that K is
not strongly affected by wear and other damage of the meter. When a rotor with high
solidity 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 to
wear or impact than rotors with lower solidity. We therefore focus on rotors with
high solidity.
Results of a two-dimensional model described in chapter 2 have been compared
to the calibration data of Elster-Instromet for two different turbine flow meter geometries. The theory assumes that the flow is similar to that in a two-dimensional cascade
with full fluid guidance. The theory predicts global trends. Small changes in rotor
blade 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 is
Reynolds number dependent (section 2.3). This appears to have a significant effect
id
on the deviation, E = ω−ω
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 the
asymmetric shape of the trailing edge of the blade. This partially explains why the
actual 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 the
pipe wall has an important effect on the deviations from ideal flow (section 2.6.3).
By placing the tip of the rotor blade in a cavity in the pipe wall the manufacturers
can obtain a rotor constant K which is almost independent of the Reynolds number.
A theoretical model of the tip clearance flow effect should therefore be included to
obtain 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 these
high frequencies the inertia of the rotor is so important that a steady rotation speed of
the rotor is achieved , so that dω
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 of
0
pulsations. This quasi-steady flow model predicts that the deviation Epuls = ω−ω
ω0
from the steady flow response
depends
quadratically on the r.m.s value of the relative
velocity amplitude Epuls =
|u′in |2
u2in
, is compared with measurements, where uin 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
We therefore
remains.
|u′in |2
. The error caused
introduce a quadratic fit parameter a such that Epuls = a u2
in
by the pulsation is dependent on Strouhal number (section 3.6.1). The quadratic fit
1
parameter, a, of the measurement errors decreases as SrL5 blade for SrLblade < 2.5. This
particular power law is not yet understood. This quadratic fit parameter is not dependent on Reynolds number (section 3.6.2). Measurements at higher Strouhal number
were not accurate enough to confirm this dependence at higher Strouhal numbers.
The deviation obtained for the superposition of two harmonic perturbations can be
predicted by addition of the deviations caused by the two individual perturbations.
Tests with a different rotor indicate that the blade shape does not affect the pulsation error, Epuls (section 3.6.4).
Determination of the velocity amplitude at the rotor is critical for correcting the
measurements. Local measurements of the velocity either by hot wire anemometers
or pressure transducers are not reliable (section 3.4.4). More global measurements
using 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 experimentally 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 rounded
leading edge and a sharp trailing edge. Experiments using a rotor with sharp trailing and leading edges show that no ghost counts occur, even when the symmetry is
broken by an initial rotation of the rotor (section 4.3.2).
An explicit equation calculating the edge force on an isolated flat plate using
potential 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 the
thickness of the rotor blade on the edge force. A comparison of the visualisation of
the flow around the edge of a blade with the discrete vortex potential flow solution
shows reasonable agreement. Similar characteristics in the pressure at the edge are
found in the results of the discrete vortex model and the experiments (section 4.4.4).
5.5
Recommendations
The aim of this study was to develop and verify simplified analytical models to be
used 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 points
in the design, but a better prediction of the Reynolds number dependence can only be
achieved by including realistic tip clearance effects in the theory.
The quasi-steady flow model predicting the measurement error during pulsating
flow provides a reasonable (within 40%) prediction for Strouhal numbers based on
the blade tip chord length, SrLblade up to 10. For Strouhal number pulsations up
to 2.5 an empirical relation (equation 3.26) can be used to correct the quasi-steady
theory for a more accurate prediction. It is recommended to extend this equation to
higher Strouhal numbers. This would involve new experiments with more accurate
acoustical velocity measurements at high frequencies.
Problem in the use of this model in practice is determining the velocity amplitude
of the pulsations at the rotor. Placing an array of microphones within the flow meter
appears to be the most promising option.
Using blades with a rounded leading edge and sharp trailing edge enhances the
steady performance of a rotor, but can result in spurious counts in purely oscillating
flow. An analytical expression derived in chapter 4 gives an order of magnitude
prediction of the onset of spurious counts in purely oscillating flow (without main
flow). Using this tool possible problems of ghost counts can be located. By using the
cascade theory of Durant (1963) this model can be extended to include interaction
between blades. Furthermore, a more accurate determination of the edge force using
a numerical method can improve the accuracy of the prediction.
APPENDIX
A
Mach number effect in temperature
measurements
Assuming that the temperature sensor placed in the meter measures a temperature
close to the adiabetic wall temperature, Tw , for a turbulent boundary layer on a flat
plate we have (Shapiro, 1953):
√ γ−1 2
Tw
≈ 1 + Pr
M ,
T∞
2
(A.1)
where Pr is the Prandtl number, γ is Poisson’s ratio and T∞ is the main flow temperature. In general the meter is calibrated against a series of three other meters placed
in parallel, so that the Mach number at the meter being calibrated is about a factor 3
higher than at the reference meters. Assuming Tw ≈ T∞ induces a calibration error
in air of the order 0.18M2 .
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 no
means a complete analysis, only some simplified boundary layer calculations used in
this thesis will be addressed.
The flow in the turbine flow meters investigated in this thesis operate at flow
conditions with high Reynolds numbers, Re = UνL , with U the main stream velocity,
L a characteristic length scale and ν the kinematic viscosity. For high Reynolds
number flows, Re >> 1, viscous forces can be neglected, except within a thin layer
near the wall. Near the wall viscous forces are dominant, causing the fluid to stick
to the wall. This thin layer can be described by boundary layer theory, while for the
bulk 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.
U
U
u ( x ,y )
d (x )
y
x
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 ∂v
+
=0,
∂x ∂y
∂u
∂u
∂u
1 ∂p
∂2u
+u
+v
=−
+ν 2 ,
∂t
∂x
∂y
ρ ∂x
∂y
1 ∂p
−
=0,
ρ ∂y
(B.1)
where the x-axis is along the wall and the y-axis is perpendicular to the wall, u is
the velocity in the x-direction and v is the velocity in the y-direction. The pressure
variation in streamwise direction in the bulk flow is imposed on the boundary layer
and can be found using the Euler equation for the bulk flow. For the steady flow along
the flat plate this leads to;
∂U (x)
∂p
= −ρU (x)
(B.2)
∂x
∂x
The thickness of this boundary layer, δ, is difficult to determine, because the velocity increases smoothly from zero at the wall and reaches asymptotically to the
free stream velocity, U , giving it no exact limit. A well-defined quantity is the displacement thickness, δ1 . The displacement thickness is the distance at which a solid
boundary would be placed in order to keep the mass flux equal to the mass flux of the
flow with boundary layers;
Z δ
Z ∞
u
u
δ1 (x) =
1−
dy =
dy ,
(B.3)
1−
U
U
0
0
where the upper limit of the integrant is extended to infinity, because for y ≥ δ u = U
and the integrand is zero.
Another useful and well-defined quantity is the momentum thickness, δ2 . The
momentum thickness is defined to account for the loss of momentum for the case
where the flow has no boundary layer, but is corrected with the displacement thickness and the actual flow;
Z ∞ u
u
1−
dy .
(B.4)
δ2 (x) =
U
U
0
For a uniform flow through a channel of height, H, the total momentum flux is equal
to ρU 2 (H − 2 (δ1 + δ2 )).
For some simple cases an exact solution of the boundary layer equations B.1 can
be 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 or numerical
methods have to be used. An example of an approximate method is the Von Kármán
integral momentum equation, this equation will be discussed in section B.3.
B.2. Blasius exact solution for boundary layer on a flat plate
B.2
119
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 the
x-axis and there is no pressure gradient. The boundary layer equations B.1 reduce to:
∂u ∂v
+
=0,
∂x ∂y
∂u
∂2u
∂u
+v
=ν 2 .
u
∂x
∂y
∂y
The boundary condition are given by the no-slip condition at the wall, at y = 0
u = v = 0, and the condition of smooth transition from the boundary layer to the
main stream velocity, at y = ∞ u = U . Blasius supposed that the dimensionless
velocity, Uu , at various distances from the edge is self-similar, i.e. depends on y/δ(x),
making it possible to make the variables non-dimensional, using the boundary layer
thickness, δ(x) and the main stream velocity, U . The boundary layer thickness based
on the solution for a suddenly accelerated plate as derived by Stokes (Schlichting,
y
1979) is found to be of the form; δ ∼ νx
U . Using η ∼ δ
η=y
r
U
,
νx
(B.5)
and introducing a stream function, ψ;
√
ψ = νxU f (η) ,
(B.6)
the second-order partial differential equation can be transformed in a third-order ordinary differential equation;
2
d3 f
d2 f
+
f
=0,
dη 3
dη 2
(B.7)
with boundary conditions;
η=0
f =0
η→∞
df
=1.
dη
and
df
=0,
dη
More details can be found in Schlichting (1979). To solve this equation analytically
is difficult and was done by Blasius using power series expansion.
120
B. Boundary layer theory
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:
r
r
∂u
U ν ′′
νU
= ρU
f (0) ≈ 0.332ρU
.
(B.8)
τw (x) = µ
∂y y=0
x
x
Using equation B.3 the displacement thickness of a boundary layer on a flat plate
becomes;
r
νx
.
(B.9)
δ1 (x) ≈ 1.7208
U
B.3
The Von Kármán integral momentum equation
As mentioned above for most problems an exact solution cannot be found. For these
problems an approximate method can be used. Integrating the momentum equation
for steady flow, a global solution can be found;
h
Z h
∂u
∂u
dU
∂2u
u
+v
−U
dy =
ν 2 dy .
∂x
∂y
dx
y=0
y=0 ∂y
Z
(B.10)
where h(x) is outside the boundary layer for all values of x.
Using the definition of shear stress, τw ;
τw = µ
∂u
|y=0 ,
∂y
(B.11)
and replacing the normal velocity component, v, using the continuity equation with;
Z y
∂u
dy ,
(B.12)
v=−
0 ∂x
equation B.10 becomes:
Z
dU
τw
∂u ∂u y ∂u
−
dy − U
.
dy = −
u
∂x ∂y 0 ∂x
dx
ρ
y=0
Z
h
(B.13)
By integrating the second term on the left-hand side by parts, the equation can be
rewritten as;
Z
h
y=0
dU
∂
(u (U − u)) dy +
∂x
dx
Z
h
y=0
(U − u) dy =
τw
.
ρ
(B.14)
B.4. Description laminar boundary layer
121
Using the definitions of displacement thickness, δ1 , (equation B.3) and momentum thickness, δ2 , (equation B.4) we get the Von Kármán equation:
dU
τw
d
U 2 δ2 + δ1 U
=
.
dx
dx
ρ
(B.15)
To solve this equation a velocity profile, Uu , has to be assumed. With this velocity profile the displacement thickness, the momentum thickness and the shear stress
at the wall can be determined and the Von Kármán equation can be solved. The
next sections will introduce an approximate laminair and turbulent boundary layer
descriptions used in this thesis.
B.4
Description laminar boundary layer
Pohlhausen (1921) used a fourth-order polynomial to describe a laminar boundary
layer, however, Hofmans (1998) and Pelorson et al. (1994) found that a third-order
polynomial gives a more accurate description of the boundary layer. The velocity in
the boundary layer can than be described as:
3
X y i
u
ai
=
.
U
δ
(B.16)
i=0
To satisfy the no-slip condition at the wall and a smooth transition to the main
stream velocity at the edge of the boundary layer the following four boundary conditions are introduced;
for
y=0:
u=0
for
y=δ:
u=U
ν
dU
∂2u
= −U
,
2
∂y
dx
∂u
=0.
∂y
Introducing the non-dimensional parameter quantity, λ =
of the polynomial can be found;
a0 = 0 ,
3 λ
a1 = + ,
2 4
λ
a2 = − ,
2
1 λ
a3 = − + .
2 4
δ 2 dU
ν dx ,
the coefficients
122
B. Boundary layer theory
Using the definitions for the displacement thickness, δ1 , (equation B.3) and the
momentum thickness, δ2 , (equation B.4), it can be found that;
1
3
− λ δ,
δ1 =
8 48
39
1
1 3
δ2 =
−
λ−
λ δ.
280 560
1680
If the pressure gradient is neglected, λ = 0, the exact solution of Blasius can be
found.
Using this third-order polynomial description the shear stress, τw , at the wall can
be calculated and is:
∂u
U
τw = µ
(B.17)
= a1 µ .
∂y y=0
δ
B.5
Description turbulent boundary layer
If fully turbulent flow is considered the velocity and pressure components can be
separated into a mean motion and a fluctuation,
u = ū + u′
v = v̄ + v ′
p = p̄ + p′ .
(B.18)
To approximate the turbulent boundary layer, we are only interested in average
velocities 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, that
the velocity profile for a turbulent flow is quite complex composition of logarithmic
function and an additional linear layer. However, for many practical applications, it
has been shown experimentally that the velocity profile can be approximated by a
simple equation in the form of a 1/7th power (Schlichting, 1979);
y 1
u
7
.
=
U
δ
(B.19)
Using this equation the displacement thickness, δ1 , and the momentum thickness, δ2
are;
1
δ1 = δ ,
8
7
δ2 = δ .
72
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 impossible
to calculate the shear stress caused by the boundary layer on the wall and a empirical
relation has to be found. In this thesis, as an approximation the shear stress found for
a fully developed turbulent pipe flow;
τw = 0.0225ρU 2
ν 1
4
Uδ
(B.20)
124
B. Boundary layer theory
C
Measurements
C.1
Introduction
The effect of velocity pulsations on flow measurements has been investigated in this
thesis by measuring this error at resonance frequencies between 24 Hz and 730 Hz
and relative velocity amplitudes, uac /u0 , ranging from about 0.01 to 2. In this appendix the result are represented for the measurements carried out at pulsation frequencies of 24, 117, 360 and 164 Hz. The data is plotted for every pulsation frequency separately using a double logarithmic scale as well as a linear scale. The
mainstream velocity at which the measurement is performed are indicated by the
different symbols.
126
C.2
C. Measurements
Pulsation frequency of 24 Hz
0
quasi−steady theory
u0 = 10 m/s
relative measurement error, Epuls
10
u = 2 m/s
0
−1
u0 = 5 m/s
10
−2
10
−3
10
−4
10
−2
−1
10
10
relative acoustic amplitude, |u’|/u
0
10
0
Figure C.1: The relative measurement error, Epuls , as a function of the relative amplitude
of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 24 Hz.
Plotted using double logarithmic scale
relative measurement error, Epuls
2
quasi−steady theory
u0 = 10 m/s
u = 2 m/s
1.5
0
u0 = 5 m/s
1
0.5
0
0
0.5
1
1.5
relative acoustic amplitude, |u’|/u
2
0
Figure C.2: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 24 Hz.
C.3. Pulsation frequency of 69 Hz
C.3
127
Pulsation frequency of 69 Hz
0
quasi−steady theory
u0 = 5 m/s
relative measurement error, Epuls
10
u = 0.5 m/s
0
−1
u = 1 m/s
10
0
u0 = 1.5 m/s
u = 2 m/s
0
−2
10
u0 = 3 m/s
−3
10
−4
10
−2
−1
10
10
relative acoustic amplitude, |u’|/u
0
10
0
Figure C.3: The relative measurement error, Epuls , as a function of the relative amplitude
of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 69 Hz.
Plotted using double logarithmic scale
relative measurement error, Epuls
2
quasi−steady theory
u0 = 5 m/s
u = 0.5 m/s
1.5
0
u = 1 m/s
0
u0 = 1.5 m/s
u = 2 m/s
1
0
u0 = 3 m/s
0.5
0
0
0.5
1
1.5
relative acoustic amplitude, |u’|/u
2
0
Figure C.4: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 69 Hz.
128
Pulsation frequency of 117 Hz
relative measurement error, E
puls
C.4
C. Measurements
quasi−steady theory
u = 3 m/s
0
−1
10
u = 1 m/s
0
u = 2 m/s
0
−2
10
−3
10
−4
10
−2
−1
10
10
relative acoustic amplitude, |u’|/u
0
10
0
Figure C.5: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 117 Hz.
Plotted using double logarithmic scale
0.5
relative measurement error, E
puls
quasi−steady theory
u = 3 m/s
0
0.4
u0 = 1 m/s
u = 2 m/s
0
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
relative acoustic amplitude, |u’|/u
1
0
Figure C.6: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 117 Hz.
C.5. Pulsation frequency of 363 Hz
C.5
129
Pulsation frequency of 363 Hz
0
10
0
puls
relative measurement error, E
quasi−steady theory
u = 15 m/s
u = 1 m/s
0
−1
10
u0 = 2 m/s
u = 5 m/s
0
−2
u = 10 m/s
0
10
−3
10
−4
10
−2
10
−1
10
relative acoustic amplitude, |u’|/u
0
10
0
Figure C.7: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 360 Hz.
Plotted using double logarithmic scale
0.5
relative measurement error, E
puls
0.45
quasi−steady theory
u = 15 m/s
0
0.4
u = 1 m/s
0.35
u0 = 2 m/s
0.3
0.25
0
u = 5 m/s
0
u = 10 m/s
0
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
relative acoustic amplitude, |u’|/u
1
0
Figure C.8: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 360 Hz.
130
Pulsation frequency of 730 Hz
relative measurement error, Epuls
C.6
C. Measurements
−1
10
quasi−steady theory
u0 = 15 m/s
u0 = 1 m/s
u0 =2 m/s
−2
10
u0 = 5 m/s
u0 = 10 m/s
−3
10
−4
10
−2
10
−1
10
relative acoustic amplitude, |u’|/u
0
10
0
Figure C.9: The relative measurement error, Epuls , as a function of the relative amplitude of
the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 730 Hz.
Plotted using double logarithmic scale
0.5
relative measurement error, Epuls
0.45
0.4
0.35
0.3
quasi−steady theory
u0 = 15 m/s
u0 = 1 m/s
u =2 m/s
0
u0 = 5 m/s
u0 = 10 m/s
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
relative acoustic amplitude, |u’|/u0
1
Figure C.10: The relative measurement error, Epuls , as a function of the relative amplitude
of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 730
Hz.
D
Force on leading edge
Using the transformation of Joukowski (see equation 4.1), the value of dξ/dz can be
calculated close to the leading (singular) edge.
lim
z→−2A
"
#
√
1
z
i A
dξ
= lim
+ p
= lim √
(D.1)
z→−2A 2
z→−2A 2 z + 2A
dz
2 (z − 2A)(z + 2A)
where A is the radius of the circle in the ξ-plane. Because near the edge (z →
dξ
becomes very large and the second term becomes the dominant term and
−2A) dz
equation D.1 is obtained. The potential of the flow is given by equation 4.2. With this
equation and the Kutta condition imposed at the trailing edge, z = 2A, (see equation
4.3) the circulation, Γv , can be found:
Γv = 4πuac sin α
A(A − ξv )(ξv∗ − A)
ξv ξv∗ − A2
(D.2)
where uac is the acoustic oscillation amplitude, α is the incidence of the flow and
ξv is the position of the vortex in the transformed plane. Using the flow potential, Φ
and the circulation Γv , dΦ/dξ close to the leading edge can be calculated:
dΦ
dξ
ξ→−A
1
ξv∗
iΓv
+
= uac e
− uac e −
2π −A − ξv
−Aξv∗ − A2
(A − ξv )(A − ξv∗ )
(D.3)
= −2iuac sin α 1 +
(A + ξv )(A + ξv∗ )
−iα
iα
By combining equations D.1 and D.3 it follows that:
132
D. Force on leading edge
dΦ
dz
z→−2A
dΦ
dξ
=
dξ ξ→−A dz z→−2A
√
(A − ξv )(A − ξv∗ )
1
√
= uac A sin α 1 +
(A + ξv )(A + ξv∗ )
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 contour ǫ
iρ
Fx − iFy =
lim
2 ǫ→0
I ǫ
dΦ(z)
dz
2
dz
(D.5)
where Fx denotes the force parallel to the plate (the edge force) and Fy represents
the force perpendicular to the plate.
Using the Cauchy integral theorem, the force becomes:
Fx = Fe
(A − ξv )(A − ξv∗ ) 2
1+
=
(A + ξv )(A + ξv∗ )
2
A2 + ξv ξv∗
2
2
= −4πρuac A sin α
(A + ξv )(A + ξv∗ )
−πρu2ac A sin2 α
(D.6)
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Summary
Dynamics of turbine flow meters
Axial turbine flow meters are used in applications, in which accurate volume flow
measurements are desired. When used at ideal conditions, turbine flow meters allow
gas flow measurements with an accuracy of about 0.2%. The aim of our research is to
develop engineering tools allowing to design robust and accurate turbine flow meters.
Ideally, the volume flow is proportional to the rotation speed. However, slight
deviations are observed as a function of the Reynolds number. In designing a flow
meter, the object is to make the dependence on the Reynolds number as small as possible and to create a robust meter, not sensitive to wear nor damage. In this thesis we
describe a two-dimensional analytical model to predict the deviation of the rotation
speed 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 with
calibration data of turbine flow meters provided by Elster-Instromet, we find that the
theory can explain global effects of the inlet velocity profile, the pressure drag of
the wake and other friction forces. For a more accurate model it is necessary to include realistic tip clearance effects and the additional lift induced by the shape of the
trailing edge.
Another important source of systematic errors are time dependent perturbations
in the flow. We investigate the effect of pulsations at high frequencies on the rotation speed of the rotor. Above a critical frequency, determined by the inertia of the
rotor, the turbine meter will not be able to follow the variations in volume flow in
time. Instead an average rotation speed will be established. Due to non-linearities of
the forces exerted by the flow on the turbine blades this rotation speed corresponds
to a higher steady volume flow than the actual time-averaged flow. By assuming
quasi-steady behaviour of the flow for the model obtained for ideal steady flow described above, the relative error in volume flow is equal to the root-mean-square of
the 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 significant
deviations from the quasi-steady model were found, the quadratic dependence on the
velocity amplitude appears to remain valid for all measurements. The exact quadratic
dependence is a function of the Strouhal number of the pulsations. In the range of
Strouhal numbers below 2.5, based on the blade chord length at the tip of the rotor
blade and the flow velocity at the rotor inlet plane, we find a slow decrease of the
error with increasing Strouhal number following a power 15 of Strouhal. Measurements at high Strouhal numbers were not reliable enough to confirm this dependence
for higher Strouhal numbers. The deviation obtained for the superposition of two
harmonic perturbations can be predicted by addition of the deviations caused by the
two 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 to
predict the onset of the ghost counts. This equation is verified by experiments to be an
engineering tool allowing to predict the order of magnitude of the onset of spurious
counts. A more accurate prediction of the onset of spurious counts can be found by
studying the flow around the rotor blades. Experiments were performed to measure
the pressure at three different locations on a scale model of the edge of a rotor blade
and to visualise the flow around this edge. These measurements and visualisations
show 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 vaak
axiale turbinedebietmeters toegepast. Onder ideale omstandigheden kunnen turbinedebietmeters voor gasstromingen een nauwkeurigheid bereiken tot 0.2%. Het doel
van dit onderzoek is om technologische applicaties te ontwikkelen die het mogelijk
maken robuuste en nauwkeurige meters te ontwerpen.
In het ideale geval is het volumedebiet rechtevenredig met de rotatiesnelheid
van de rotor. In praktijk echter treden er kleine afwijkingen op als functie van
het Reynoldsgetal. Bij het ontwerpen van een turbinemeter is het doel om deze
afhankelijkheid van het Reynoldsgetal zo klein mogelijk te maken. Tegelijkertijd
moet de turbinemeter ook robuust zijn, zodat hij niet gevoelig is voor slijtage en
kleine beschadigingen. In dit proefschrift wordt een twee-dimensionaal analytisch
model beschreven dat het mogelijk maakt afwijkingen te voorspellen van de rotatiesnelheid van de rotor met de rotatiesnelheid van een ideale rotor, waarbij de bladen
de vorm hebben van een ideale helix, die ronddraait zonder mechanische wrijving
of stromingswrijving. De resultaten van dit model zijn vergeleken met ijkmetingen
die beschikbaar zijn gesteld door Elster-Instromet. Uit deze vergelijking kunnen we
concluderen dat het model de globale effecten van het instroomprofiel, van de drukweerstand van het zog en van andere wrijvingskrachten verklaart. Om de invloed
van de ruimte tussen de uiteinden van de rotorbladen en de wand te verklaren is een
uitgebreider model nodig. Dit geldt ook voor het effect van de additionele liftkracht
die opgewekt wordt door de vorm van de achterkant van de rotorbladen.
Een andere belangrijke bron van systematische fouten zijn tijdsafhankelijke verstoringen in de stroming. Het effect van pulsaties met hoge frequenties op de rotatiesnelheid van de rotor is onderzocht. Boven een kritische frequentie, die bepaald wordt
door de massatraagheid van de rotor, is de rotor niet meer in staat de variaties in het
volumedebiet te volgen. In plaats hiervan stelt zich een gemiddelde rotatiesnelheid
140
Samenvatting
in. Door niet-lineariteit van de krachten die door de stroming worden uitgeoefend
op de turbinebladen, correspondeert deze rotatiesnelheid met een volume debiet dat
groter is dan het tijdsgemiddelde debiet. Door aan te nemen dat het gedrag van de
stroming quasi-stationair is en door gebruik te maken van het model dat hierboven
is beschreven, vinden we dat de fout in het gemeten volumedebiet gelijk is aan de
rms-waarde van de verhouding tussen de akoestische snelheid en de hoofdstroomsnelheid. We hebben de grenzen onderzocht waarbinnen dit model voor harmonische
pulsaties geldig is. Alhoewel we significante afwijkingen van de quasi-stationaire
theorie hebben gevonden, lijkt de kwadratische afhankelijkheid van de snelheidsverhouding geldig te blijven. De exacte kwadratische afhankelijkheid is een functie van
het Strouhalgetal. Voor Strouhalgetallen, gebaseerd op de koorde van de bladen aan
de uiteinden en de snelheid bij het binnengaan van de rotor, kleiner dan 2.5, vinden
we een langzame afname van de fout met toenemend Strouhalgetal met een macht 51
van het Strouhalgetal. Metingen voor hogere Strouhalgetallen bleken niet voldoende
betrouwbaar om deze trend te bevestigen voor hogere Strouhalgetallen. De afwijking
voor twee gesuperponeerde harmonische verstoringen kan voorspeld worden door de
afwijkingen van de twee afzonderlijke verstoringen bij elkaar op te tellen.
Voor extreme situaties waar pulsaties optreden, maar er geen tijdsgemiddelde
stroming is, is het mogelijk dat de rotor begint te roteren als er pulsaties zijn met
een amplitude hoger dan een kritische amplitude. Deze foutieve tellingen of ”spooktellingen” worden veroorzaakt door de vorm van de rotorbladen. Door gebruik te
maken van potentiaaltheorie voor een stroming om een oneindig dunne rotorblad,
kan een expliciete uitdrukking gevonden worden, die de aanvang van spooktellingen kan voorspellen. Door middel van experimenten is deze uitdrukking geverifiëerd
en kan hij worden toegepast om een orde-grootte-voorspelling te doen voor de aanvang van foutieve tellingen. Een nauwkeurigere voorspelling kan gevonden worden
door de stroming om een rotorblad te bestuderen. Een experimentele opstelling is
gebouwd om de druk op drie verschillende locaties op een schaalmodel van de rand
van een rotorblad te meten en om de stroming rondom de rand te visualiseren. Deze
metingen en visualisaties laten een redelijke overeenkomst zien met de resultaten van
een discreet wervel-model.
Dankwoord
Het is natuurlijk een cliché te schrijven dat het proefschrift dat hier ligt niet het werk
is van alleen één 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 bijdrage
hebben geleverd aan de totstandkoming van dit proefschrift.
In de eerste plaatst mijn promotor Mico Hirschberg.
Mico, bedankt voor je begeleiding, je creativiteit en al je
ideeën waarmee je me vaak hebt gemotiveerd en soms tot
waanzin hebt gedreven. Met erg veel plezier heb ik al die
jaren met je samengewerkt. Speciaal voor jou wilde ik
dit proefschrift niet afsluiten zonder een varken.
Harry Hoeijmakers, mijn 2de promotor, bedankt voor
het het grondig lezen van het concept. Dank je wel, Rini
van Dongen; voor je interesse en je raad gedurende de afgelopen jaren.
Het project waar dit proefschrift uit is voort gekomen is gefinancierd door STW.
Ik wil dan ook graag de gebruikerscommissie bedanken voor haar aandacht en input.
In het bijzonder Jos Bergervoet van Elster-Instromet. Jos, bedankt voor je medewerking en voor het delen van je jarenlange ervaring met turbine debietmeters. Ook wil
ik graag Henk Riezebos bedanken voor zijn medewerking en René Peters, die mij de
gelegenheid heeft gegeven om twee maanden bij TNO te komen werken. Het waren
een leuke en leerzame twee maanden. Stefan Belfroid, dank je wel; twee blobbers
weten meer dan één.
Ik ben erg dankbaar voor de technische steun die ik heb gekregen. Mijn gebrek
aan experimentele ervaring is uitstekend opgevangen door de jarenlange ervaring van
Jan Willems. Bedankt Jan, het was erg prettig gebruik te kunnen maken van al je
kennis, 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 computers
had van alle opstellingen. Ad Holten, bedankt voor je hulp als één van al deze com-
142
Dankwoord
puters raar deed en Freek niet aanwezig was en voor je hulp met de optica. Ook wil
ik Remi Zorge en Herman Koolmees bedanken voor hun medewerking. Daarnaast
Gerald Oerlemans; bedankt dat je bereid was alle duct tape te trotseren, toen andere
technici niet beschikbaar waren.
Natuurlijk ben ik ook het secretariaat dank verschuldigd voor het helpen met het
administratieve werk. Dank je wel, Brigitte. Merci, Marjan; ik heb erg veel plezier
gehad 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 Küchel, Arjen Hamelinck, Erwin
Engelaar, Martijn de Greef, Bram van Gessel, Floor Souren en Ineke Wijnheijmer.
De afgelopen (ruim) vier jaar waren niet zo leuk geweest zonder het gezelschap
van al mijn collega’s en oud-collega’s. Het is waarschijnlijk niemand echt ontgaan dat
ik een liefhebber ben van koffiepauzes (of eigenlijk theepauzes), met name vanwege
de discussies en gezelligheid tijdens die pauzes. Dank je wel, Marleen, Werner, Gerben, Geert, Ralph, Gabriel, Jieheng, Vincent, Dima, Laurens, Thijs, Ruben, Rudie,
Rinie, Lorenzo, Andrzej, Alejandro, Matı́as, Jurriën, Daniel, Dennis, John, David,
Moasheng, Paul, Elke, Gert Jan, Willem, Herman, Gerard, Gert en de personen die
al eerder genoemd zijn.
Vrienden en familie zijn voor mij erg belangrijk geweest. Niet alleen vanwege
de interesse die zij hebben getoond in mijn onderzoek, maar ook vanwege de nodige
afleiding die ze me hebben gegeven. Dank jullie wel, allemaal. In het bijzonder
Saskia, 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 geweest en
degene die het meest geleden heeft naast mij als het onderzoek niet goed ging. Lieve
Patrick, 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 Lungs
awarded 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.