The development of a microanemometer : new possibilities for
measuring very low air velocities
Citation for published version (APA):
Pluijm, M. J. F. P. (1987). The development of a microanemometer : new possibilities for measuring very low air
velocities. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR274691
DOI:
10.6100/IR274691
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Published: 01/01/1987
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THE DEVELOPMENT OF A MICROANEMOMETER
New possibilities for measuring very low air veloeities
THE DEVELOPMENT OF A MICROANEMOMETER
New possibilities for measuring very low air veloeities
proefschrift
Ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof. dr. F. N. Hooge, voor een commissie aangewezen door het college van dekanen in het openbaar te verdedigen op
vrijdag 11 december 1987 te 16.00 uur
door
MARTINUS JOHANNES FRANCISCOS PETRUS PLUIJM
Dit proefschrift is goedgekeurd door de promotoren
Prof. dr. J.A. Poulis en
Prof. ir. j. Vorenkamp
Aan CherryL
TABLE OF CDN'I"I!lUS
1 . GENERAL INTRODUeTION. . . . . ... 1
2. LOW-AIR-VELOCITY MEASURING PRINCIPLES ... 3
2.1 Introduetion
2.2 Review of low-air-velocity measuring principles
2.2.1 The Pitot static tube
2.2.2 Cup, vane and propeller anemometers 2.2.3 Hot-wire anemometers
2.2.4 Thermal marker and iontracing 2.2.5 The Sonic Anemometer
2.2.6 The Laser Doppier Anemometer 2.3 Conclusions and remarks
4
6 7 8
10
3. MEASURING PRINCIPLE OF THE MICROANEMOMETER ... ll 3.1 Introduetion
3.2 The measuring principle 3.3 The microanemometer
3.3.1 The optica! detection system 3.3.2 The feedback system
3.3.3 The moving-coil meter
3.4 A first comparison of the usefulness of different moving-coil meters as microanemometers
3.5 General considerations
4. CALIBRATION UNIT FOR MICROANEMOMETERS AT VERY LOW AIR
12 16 19 20 21 24 VELOeiTIES ... 26 4.1 Introduetion
4.2 The calibration unit for microanemometers 4.3 Automation of the experiments
4.4 The characteristic calibration measurement 4.4.1 Supplementary calibrations
4.4.2 A characteristic calibration 4.5 Experiments and discussion
28 30
31
33
5. NUMERICAL ANM..YSIS OF 11fE FLOW AROUND 11fE MICROANEMOMETER ... 36 5.1 Introduetion
5.2 The Navier Stokes equations 5.3 The Finite Element Metbod
5.3.1 Broad deseription of the metbod 5.3.2 Mathematica! formulation
37
39
5.4 Numerical results 43
5.4.1 Practical aspects and preliminary calculations
5.4.2 Velocity profiles and verification 45
5.4.3 Influence of the geometry of the cylinder 50
5.5 Conclusions and discussion 52
6 MEASUREMENTS WITH THE MICROANEMOMETER ... 53 6.1 Introduetion
6.2 Determining the microanemometer velocity range 6.3 Calibration measurements with different types of
microanemometers
6.3.1 Microanemometers with different pointer lengtbs
56
6.3.2 The influenee of the sphere 60
6.4 Comparison with literature: Cd vs. Reynolds curves 61
6.5 A first comparison with numerical results 64
6.6 Discussion 66
6. 7 The dynamic behaviour of the microanemometer 68
6. 7.1 Theory 6.7.2 Experiments
6.8 Discussion and coneluding remarks 71
7 MICROANEMOMETERS FOR MEASURING MULTIDIMENSIONM.. AIR VELOeifriES 72 7.1 Introduetion
7.2 Direetional sensitivity 7.2.1 Introduetion
7.2.2 Experiments todetermine G(~)
7.2.3 Experiments todetermine B(U.P)
73
75
7.3 The two-dimensional microanemometers: T2-90 and T3-120 79 7.3.1 Introduetion and definitions
7.3.2 Calibrations 82
7.3.3 Discussion 85
7.4 The three-dimensional microanemometers: D3-90 and 04-109 86 7.4.1 Introduetion and definitions
7.4.2 Numerical predictions of the D3-90 and the D4-109 88
7.4.3 Discussion 89
7.5 General considerations 92
8 MEASUREMENTS IN PRACTICE ...
9381. Introduetion
8.2 Measurements in our calibration unit determination of wall effects
8.2.1 Experiments and results 8.2.2 Discussion
8.3 Measurements in laboratory fume hoods 8.3.1 Introduetion
8.3.2 Description of the experiments 8.3.3 Experimental results
8.3.4 Conclusions and discussion
8.4 Air veloeities in a surgical operating theatre 8.4.1 Introduetion
8.4.2 Description of the experiments 8.4.3 Experimental results 8.4.4 Discussion RE FE RENCES SUMMARY SAMENVATTING NAWOORD a.JRRICULUM VITAE 96 97 98 100 102 104 106 109 110 114 115 116 118
1. GENERAL INTRODUCTION
In 1984 the project ' Development of a microanemometer was started as a cooperation between the facul ties of Physics and Architecture, Building and Planning at the Eindhoven University of Technology. lts main objective was the development of a new cheap type of microanemometer to enable measurements of air veloeities of the order of 1 cm/s up to 10 cm/s to be made. This implied lowering the velocity range of the commonly available microanemometers by one decade.
Future applications of the microanemometer were focussed on our cooperation with the department of Building Engineering. A number of possible applications are given below.
*
Measuring low air veloeities is of importance in airconditioned spaces, such as surgical operating theatres, laborator i es and clean rooms for microtechnics (Finck '78). In operating theatres and their environment several zones of sterility have to be maintained. Technically it is not possible to separate these zones by walls. Air movements are caused by the pressure hierarchy: the highest pressure should therefore be where the highest sterility is required. Air movements are minimal and mainly occur in the vicinity of the doors.*
Many physical measurements are carried out in rooms where vent i lation is minimised (laboratories, balance cases). Investigations of the influence of low air veloeities on weighing in balance cases have been 1 i mi ted to theoret i cal approaches (Massen et al. '86), because of the lack of adequate equipment to measure such air velocities.*
Investigation of the indoor elimate in rooms with specific requirements (Lammers et al. '84). To study the influence of indoor elimate on human performance, research is restricted to working situations without extreme physical conditions with their inherent physiological consequences (such as high noise levels or high temperatures). This research is focussed on office and education situations where, in cooperation with scientists fromother disciplines, such as psychology and physiology, the influence of illumination, acoustics, air and body temperature, humidi ty and air veloei ties on human comfort is investigated. Special attention is paid to undesired local cooling of the body, draught, which is mostly determined by the frequency of variations of the air velocity {Olesen '79, Olesen '85).
*
In micrometeorological research (Desjardin et al. '86) and as a device for detection of dangerous gases in coal and hard rock mines (Skinner et al. '82) there is need for low-air-velocity measuring devices.Apart from these applications the challenge of extending the lowest detectable air velocity was a stimulating enough reason for starting the project.
The investigations described in this thesis can be swmnarised as follows:
In chapter 2 the existing measuring principles are reviewed with their possibilities, advantages and disadvantages.
As these principles are more or less unsuited to our purposes the new measuring principle, developed to meet the special demands of chapter 1, will be discussed in chapter 3.
In order to be able to de termine the lowest air veloei ty ~ which could be detected with our new type of microanemometer, a calibration unit was built and is described in chapter 4.
In order to be able to investigate the disturbance by the microanemometer on the flowfield in the calibration unit the veloeities in the vicinity of the microanemometer are calculated in chapter 5, using the finite element method.
The calibration of the microanemometer wi tb the calibration unit and the influence of the dimensions of the microanemometer on the calibration is shown in chapter 6.
In chapter 7 a microanemometer is presented which is able to measure simultaneously several velocity components.
This thesis is concluded with chapter 8 in which some application measurements with the developed microanemometer are described.
2. IJJrf-AIR-VELOCITY MEASURING PRINCIPLES
2.1 INTRODUeTION
The literature describes several ways of detecting low air velocities. A possible approach to dividing anemometry into categories is given by Lilienfeld et al. '67 :
1. Instruments basedon the utilization of the kinetic energy of the gas stream. Representative devices of this type are the Pitot tube or the cup and propeller anemometers.
2. Instruments depending on the conductive-convective transfer of thermal energy from a heat souree to the gas flow. The hot-wire anemometer is a typical example of this group.
3. Instruments based on the tracer technique where the time interval between the upstream injection of a tracer and its downstream detection at a known distance is measured.
4. Instruments with which veloeities are determined from changes produced in the characterics of waves propagating within the moving medium {e.g. acoustic anemometers or Laser Doppier Anemometers) .
It should be noted bere that in this enumeration, several types of anemometers, such as the corona and glow-discharge anemometers (Desai
&
Johnston '71) and the long-range Laser Doppier Anemometers {Danielsson & Pike '83) are not taken into consideration. As their minimum detectable air velocity is of the order of approx. one m/s they are therefore not suitable for our purposes. We should mention also that the above mentioned groups can showsome overlap (e.g. the pulsed hot-wire anemometer).In the next paragraph a review of some of the above mentioned measuring principles will be given in a quasi bistorical approach.
2. 2 REVIEW OF LOW-AIR-VELOCI1Y MEASURING PRINCIPLFS
2.2.1 The Pitot static tube
One of the first air velocity meters ever designed was the Pitot static tube, named after its inventor, Henri Pitot (1695 - 1771).
For reliable use, the direction of the velocity vector bas to be known with sufficient accuracy before taking any measurements. Its operation principle is based on Bernoulli 's law applied to a one-dimensional incompressible frictionleas flow. The air velocity can be determined from the density of the air p and measuring the
pressure difference between the stagnation- and the static
pressure. Because p is accurately known in most situations, the errors can be ascribed to inaccurate measurement of the pressure difference. Discrepancies between the real value and the measured value of this difference are due to (Doebelin '83):
*
misalignment of the tube axisand
the velocity vector,*
nonzero tube diameter,*
disturbance of the velocity profile by the stem,*
the effect of viscosity. The assumption tbat the flow is frictionleas is no langer valid at lower Reynolds numbers. At sufficiently low Re numbers the viscosity of the flow exerts an additional force.Generally the Pi tot static tube is a reliable instrument for measuring static flow veloei ties ranging from 10 cm/s - 30 m/s {Finck '78).
2.2.2 Cup. vane and propeller anemometers
The use of the kinetic energy of a gas flow and transforming it into a form of useful energy bas been a widely appreciated principle for ages with a great range of applications, varying from windmill to sailing ship. It is therefore not surprisihg tbat on the basis of this principle, a series of maasurement devices bas been developed, such as cup, vane
and
propeller anemometers.*
A CUP ANEMOMETER is characterised by a number of half-spheres mounted on a vertical rotation axis. The difference in air resistance between the convex and concave sides will cause the axis to rota te. The rotation veloei ty is a measure of the air velocity. However, this difference in air resistance only applies to Reynolds numbers, Re, greater than 100, where Re is based on the diameter of the cup. In the case of a sphere wi th diameter 3 cm, this corresponds to air veloeities>
5 cm/s (Lindley '75).*
VANE and PROPELLER ANEMOMETERS include a range of simple, mechanica! and portable anemometers. One type employs a spring to resist the aerodynamic force, the other type allows vanes to whirl unobstructed as fast as is required for the net torque produced by the aerodynamic force to become zero. The first type responds to the square of the velocity, while the second has an output varying linear ly wi th the veloei ty. Di sadvantages of these sys tems are their integrating nature and a lower detection limit of approx. 20 cm/s (Leak '66), caused by friction of the journal hearing. They are therefore used for measuring mean air veloeities averaged over a langer period. As regards the limited sensitivity it should be noted here that, as an exception, Desjardin et al. '86 report a lower limit of 6 cm/s.In the case of highly varying air velocities, these systems show a tendency to overestimate the average value. For a fluctuating air velocity with a frequency of 2 Hz and an amplitude of 50% of the average value, this overestimation exceeds 10% (Finck '78).
We should at this point mention the ION DRIFT ANEMOMETERS. They drew strong attention during the
detection limit of a few cm/s
70's because of their low (Durbin et. al '71. Kurz &
Olin '71). However for reasans unknown to us no further attention was paid to them afterwards.
2.2.3 Hot-wire anemometers
Publ ications which appeared in 1900 presented a new type of
low-air-veloci ty measuring device: the hot-wire anemometer
(H.W.A.). Its use was at first limited to measuring stea.Ciy air veloeities (Comte-Bellot '76), but in later years eropbasis shifted to the maasurement of fluctuating velocities.
Hot-wire anemometry is based on the fact that the electrical reststance of a roetal conductor is a function of its temperature. The sensors are thin metallic elements heated by an electric current (Joule effect) which are caoled by the incident airflow. From the temperature (Constant Current H.W.A.) or reststance (Constant Temperature H.W.A.) attained by the sensor, it is possible to deduce information on the flow (Bestion '83). By using more than one sensor, measurements of multidimensional flows can be performed (Andreopoulos '83).
The popularity of the H.W.A. is due to the following advantages of the device (Smol'yakov '83):
*
The sensor is small enough not to introduce much disturbance andit bas good spattal resolution.
*
The response time is short so that very-high-frequency eftects (up to a few kHz) can be recorded.*
The electrical signal produced can be readily processed both by analog and digital systems.There are, however disadvantages:
*
Calibration is necessary and requires much care. Thiscalibration is very sensitive to dirtcollection and needs! to be repeated regularly (Bruun '79).
*
There are deviations from the 'eosine law' which are due to the cooling by the velocity component parallel to the wire(Bremhorst
&
Gilmore '78).*
The heat exchange mechanism is dependent upon the composi tion of the air (Andrews et al . '75).2.2.4 Thermal marker and iontracing
A measurement teclmique which is specially developed for highly turbulent flows, including regions in which reversals of the flow direction occur, has been deduced from the H.W.A. and is called the TIIERMAL MARKER (Bradbury '76, Castra & Cheun '82). It is based on the pulsed heating of the flow by a thin metal wire. A second wire, which acts as a receiving element, is placed further downstream of the flow. The velocity is determined from the interval of time between the emission by the thermal marker and the detection by the receiving wire. The factors which determine the accuracy of the velocity measurements are the probe distance, the transit time, the duration of the pulse and the fidelity with which the gas molecules follow the air.
The time constant that limits the precision of measurements on high-frequency fluctuations depends on the mimimal time interval between two subsequent heat pulses. In practice this means that the time constant of this method is greater by one order of magnitude than that of a hot-wire anemometer.
The velocity range of the thermal marker starts at approx. 15 to
20 cm/s (Kielbasa & Rysz '81, Skinner et al. '82, Westphal et al. '81).
During the last decades, in addition to heat, a variety of tracers has been used, from smoke to radioactive gases. IONTRACING, a typical memher of this group, is characterised by the fact that its measurements of the velocity are not influenced by changes in air pressure, temperature and composi ti on. Lil ienfeld et al. '67 report a veloei ty range of 0. 5 - 250 m/s wi th an accuracy of better than 5% over the entire range.
2.2.5 The Sonic Anemometer
The sonic anemometer is based on measuring the changes produced in the characteristics of waves propagating within the air. lts features include linear dynamic response, good directional characteristics and a frequency response limi ted only by the sound-path length (Coppin & Taylor '83). Several ways are reported (Kaimal & Businger '63, Mitsuta & Asai '66) of obtaining an expression for the air veloei ty, using the difference of the travel times, taking the difference of the inverse of the travel times, or using a phase-locked-loop circuit (Larson et al. '79). A resolution of a few mm/s is generally reported.
g.2.6 The Laser Dowler Anemometer
The first use of a laser Doppier anemometer (L.D.A.) to measure gas veloei ties was reported in 1964, when a lowest veloei ty of appox. 2 cm/s was detected. (Foreman et al. '65).
The operating principle of a L.D.A. relies on the presence of optica! inhomogeneities or foreign particles present in the flow or especially introduced into it. A laser beam is focussed at the point where the velocity is to be measured and a photodetector is used to detect the light scattered by particles transported by the fluid. The velocity of the particles, which is assumed to be equal to the air velocity, causes a Doppier shift of the scattered 11gbt's frequency. This can be measured with a photodetector, whose signa! is directly related to velocity. Artificial. tracer particles are not always necessary: the microscopie particles normally present in liquids or gases may suffice. It should be noted bere that experiments are mostly carried out in fluids (Yeh '64).
The advantages of the L.D.A. are: * The measurement of velocity is direct.
*
No physical object need be inserted into the flow, thus the flow is undisturbed by the measurement.*
The sensing volume can be very smal! (0.2 mmE3).* A very high frequency response (up to the MHz range) is possible.
Against this must be put the following disadvantages: * L.D.A. can only be performed in transparent tubes.
*Tracer particles in the air are required. However, in many applications seeding with particles may not be possible and may disturb the air velocity.
* The apparatus is complex and costly.
Sirnul taneous measurements of several veloei ty components at a point may be achieved ei ther by polarization schemes using a single laser (Bahnen et al. '85) or two-colour systems employing two lasers of different wavelength {Nakatani et al. '85).
2.3 CX:>NO..USIONS AND REMARKS
Several measuring principles have been described in the previous chapter. Our purpose was not to attempt completeness, but just to give the reader an impression of some of the measuring principles and their advantages and disadvantages. In order to be able to compare the techniques described above, a number of standards can be applied:
*
sensitivity to other characterics of the stream, such as pressure, temperature and humidity,*
the number of simultaneously measurable velocity components,*
spattal resolution,*
frequency response.*
complexity of the signal processing,*
cost of the apparatus,*
manufacturing complexity,*
reproducibility and accuracy,*
velocity range and more specifically the lower detection limit,*
calibration requirements.Needless to say. that depending on the application of the
anemometer, there are several more important demands. However, we rnay briefly say that all the rnethods described above are more or less unsuited to our purposes. Therefore, in consultation withand at the request of the Facul ty of Archi tecture, Building and
Planning a new measuring principle was developed and tested, which will be discussed in th~ next chapter.
3 MEASURING PRINCIPLE
OF TIJE
MICROANEMOMETER3.1
INTRODUeTION
In the previous chapter several measuring principles were described which, however were not really suited for the purposes described in chapter 1. In the present chapter a new measuring principle, developed especially to meet the specific demands will be introduced. It is based upon the measurement of the force (or rather the moment of force) exerted on an object placed in an airstream. For this force measurement a compensation balance methad is used to prevent displacements of the object. Such displacements could influence the measured force and the compensating force can be determined with precision.
The measuring device includes a balance, consisting mainly of an ordinary moving-coil meter. By sending a current through the coil, the Lorentz couple can serve as retroactive couple. In order to put the beam in its original position (which is detected by an optica! detection system), the magnitude of the Lorentz couple (the current) is adjusted with a feedback system. In that case the feedback current is a measure of the air velocity. The measuring principle is shown schematically in fig. 3.1.
air velocity
t
Mlorentz
Vout I
L J
Fig. 3.1 Schematic representation of the measuring principLe of
3.2 TIIE MEASURING PRINCIPLE
The equation of motion of the microanemometer, consisting mainly of the moving-coil meter. is
..
.
J
a(t) + K a(t) + C a(t)=
M(t) - G I(t) 3.1where J is the moment of inertia of the balance with respect to its rotation axis,
K the damping constant, C the torsion constant,
a the angle of rotation,
M the moment of force. due to the force by the airstream exerted on the measuring object,
I the current through the coil and
G a constant concerning the Lorentz couple, which we shall call sensitivity constant.
When compensation is achieved by the feedback system,
..
.
a(t) = 0, a(t) = 0 and a(t)
=
0, eqn. 3.1 becomesG
=
M/I 3.2Hence the value of M can be calculated from the value of I, assuming the value of G is known. G bas to be determined by a calibration experiment. which can be done in either of two ways.
*
The first way is based on the use of the microanemometer with feedback compensation. When applying eqn. 3.2 in calibration M hasto be known. Our first attempts were aimed at calibration of the anemometer in the position in which its rotation axis was vertical. This implied the need to apply a small horizontal force. It proved to be unpractical to use this method.
Success was achieved using the apparatus in such a way that i ts rotation axis was placed horizontally. This enabled us to provide the couple M by suspending small weights, mass m, from the end of the pointer {also being in a horizontal posi tion) and measuring the equilibrium values of
I. M
satisfiesM=mgL 3.3
where L is the lengthof the pointer of the moving-coil meter.
By plotting I vs. M. a linear relationship is obtained so that G can be determined from the slope of the line. Results are shown in
fig. 3.2. 24r---~--·
T
16 I (!lA) 8 0.15 0.30 0.45 M (j.!Nm) - >Fig. 3.2: A plot of I vs M to determtne the sensitivity constant G
This metbod results in a direct measurement of G and is preferabie when a complete microanemometer has been constructed, including the optica! detection system and the electronic circuitry.
*
The second, indirect way to determine G is consirlering the Lorentz couplewhen a current is passed through the coil. This can be written asG I
B I A Nc c c c
where B is the magnet ie induction, I the current through the coil,
c
A the area of a coil winding and c
N the number of windings. c
For G this yields
G =BA N
c c
3.4
3.5
In practice it is impossible todetermine the three constants of eqn. 3.5 (B, Ac and Ne} without irreversible damage to the moving-coil meter. When consirlering the moving-coil meter as it is commonly used as ammeter or voltmeter the Lorentz couple is compensated by the spiral springs. This leads to
G I
c -Ca 3.6
The calibration of G thus involves the measurement of a, Ie and C.
a/Ie can easily be obtained by measuring afs/Ifs , using the full-scale deflection afs and full-scale current Ifs of the meter.
The torsion constant C can be determined in a similar way to the sensitivity constant G, but now in absence of the feedback system. The microanemometer is placed so that its axis and the pointer are horizontal. Small weights, mass m, are suspended from the end of
By platting a versus M, a linear relationship is obtained and C
can be determined from the slope of the line. Here M is given by
M = m g L cos a
The results are shown in fig. 3.3
0.9.---.---.---.-~----,
T
0.6 a (rad.) 0.3 0.4 0.8 1.2 1.6 M (J,I.Nm) -3.7Ftg. 3.3 A plot of a vs M to determine the torsion constant C
It is worth while mentioning bere that this procedure is more convenient for several purposes than the earlier mentioned direct calibration of G. The procedure becomes specially attractive when different moving-coil meters, chosen out of a set, have to be compared for future use. That is because this procedure avoids the time consuming manufacture of a complete prototype microanemometer with optical detection sytem and feedback circuit.
In eqn. 3.1 there are two more constants,
J
and K, determining the mechanica! properties of the microanemometer. However, insight into these two constants is of minor importance bere. A further discussion is given in chapter 6.3. 3 THE MICROANEMOMETER
The three basic parts of the microanemometer are 1. the optical detection system,
2. the feedback system and 3. the moving-coil meter.
3.3.1 The optica! detection system
During all experiments the same simple, usefull optica! detection system was used. This optical detection system consists of a small strip of me tal attached to the pointer. a light emi tting diode (L.E.D.) and two photodiodes, placed side by side, see fig. 3.4. The two photodiodes used are rectangular silicon photodiodes Siemens BPW 34. An infrared gallium arsenide L.E.D. which reasonably matebed the photodiodes was chosen.
1. Photodiodes 2. Metal Strip 3. L.E.D. 4. Pointer
Ftg. 3.4 The opttcal detectton system
The strip is positioned so that, in the equilibrium posi tion of the meter. the shadow of the strip covers the two photodiodes equally. These photodiodes are circuited with an operational amplifier in such a way that the latter's output signal is related
to the difference of the currents of the photodiodes. Thus, when the pointer starts to leave the equilibrium position this causes the output of the operational amplifier to differ from zero.
As regards the geometry and the mechanica! construction of the optical detection system, special attention was paid to two aspects.
*
First the influence of the width of the metal strip (d} on the relation between the output current, Iph' of the photodiodes and a was studied. The results are shown in fig. 3.5.It is found that the width of the strip is of minor importance. As long as the width is not chosen greater than approx. 4 mm. the relationship between Iph and a is linear over quite a wide deflection range.
a
-0.01 rad.
Fig. 3.5 The output current Iph' of the photodiodes vs. a, the angte of deflectf.on, for severa1. widths, d, of the metal strip
M Second the influence of the distance, h, between the strip and
the photodiodes on the relationship between Iph
and
a is studied. In fig. 3.6 the results of Iph vs a are shown for h=
0.5, 1 and 2 mm and d=
3.0 mm. 0.01rad.
0.5 1.0 2.0 a-Fig. 3.6 Iph vs. a for three dtstances between strip and
photodiodes, h: 0.5, 1 and 2 mm (d
=
3.0 mm)In fig. 3.6 i t is shown that, for distances up to 2 mm, the influence of the distance between the strip and the photodi~des on the relation between Iph and a is negligible.
Summarising, we can say tbat no special care bas to be taken in this respect during the mechanica! construction of the optica! detection system.
3.3.2 The feedback system
A
Proportional Integral Compensator was chosen as feedback circuit. In fig. 3.7 the complete circuitry of the feedback system is given.photodiodes
R .
ga1nFig. 3.7 Feedback circuitry
R R
c
IL
coil 4---R sAs regards the choice of this feedback circuit, it should be noted here that we are well aware of the fact that the feedback circuit used may not be the optimum solution. However, the simplicity of the circuit and the fact that it works without any complications makes the choice acceptable.
A P-I compensator is characterised by two parameters: the time constant t
1 and the gain P:
t
1 is chosen, according to the 11'/4 phase-margin cri terion (see,
for instance, d'Azzo
&
Houpis '66 or Banks '86).The optimum value of P is determined experimentally. When the value of the gain was chosen too high the pointer started to oscillate. In each construction the optima! value of the gain is determined by increasing P until oscillations occur and then decreasing this value slightly. The resulting sensitivity was always sufficient for our purposes.
3.3.3 The moving-coil meter
A dozen different types of such meters were tested for possible use as microanemometers. These types differed in make, full-scale deflection and specified accuracy. The construction of a microanemometer with a moving-coil meter is shown in fig. 3.8
-\::->'\---magnet coil
magnet co it
Fig. 3.8
A
schematic representation of the microanemometerThe microanemometer shown in fig. 3.8 consists mainly of a moving-coil meter. The original pointer of the moving-coil meter is used as force-measuring device. To complete our microanemometer a hollow aluminium sphere is mounted round the greater part of the moving-coil meter used. A hole in this sphere allows the pointer
3.4 A FIRST OOMPARISDN OF THE USEFULNESS OF DIFFERENT
MOVING-OOIL METERS AS MICROANEMOMETERS
First applicability aspects are discussed of a given moving-coil meter which can readily be predicted without manufacturing a complete microanemometer by which is meant that only G has to be determined and that the full-scale current and the class of the instrument are lmown. This prediction allows us to carry out a preselection in a great number of available moving-coil meters, so that the time-consuming manufacture and calibration experiments can be reduced to a minimum.
For an actual preselection the values of Umi are required. the lowest detectable air velocity which we define by the S.N.R. (Sigrml to Noise Ratio) being equal to unity. Unfortunately this U mi is not a readily measurable quant i ty in the sense described above. A quantity which can however be determined is Ucl' calculated from the class of the moving coil meter specified by the manufacturers. It should be noted here that we are well aware of the fact that this quantity, to be defined later on, may well be larger by an appreciable factor than umi'
To define Ucl more precisely, the moving-coil meter is considered again. From the specifications the full-scale current Ifs and the class Cl (usually given in percentages) are used. The quantity Iel is defined by
3.8
With eqn. 3.2, Mfs and Mei are defined respectively by:
3.9
In order to introduce Ufs and Ucl we assume the relation between M and U to satisfy
3.11
where p is the density of air,
De the diameter of the cylindrical pointer, L
0 the length of the part of the pointer inside the sphere, U the magnitude of the air velocity and
cd the drag coeffient.
The drag coefficient cd satisfies c2
Cd= c 1 Ree 3.12
where c1 and c
2 are constauts which can be determined from
literature by linearisation of the double logarithmic Cd vs Ree curve and Ree is the Reynoldsnumber, based on the diameter of the cylindrical pointer
Re = U•D /v 3.13
c c
where v is the kinematic viscosity of air.
Taking M
=
Mfs and U=
Ufs in eqn. 3.11 leads to a relation between Mfs and Ufs which we consider as the definition of Ufs·By
a similar procedurewedefine Ucl as a function of Mcl·In table 3.1 we give an excerpt of the preselection results for four moving-coil meters, three of which were considered to be good enough for further development.
Several conclusions can be drawn from table 3.1. First of all we consider the conclusions which are relevant for the minimum air velocity
ucl.
Type Ia Ib Ie II
diam. magnet (mm) 21 21 21 55
G (10E-4 Nm/A} 44 lSO 300 40
Iel (J.IÁ.} 20 2 1 15 Ifs (mA) 1 0.1 0.05 1 L (mm} 70 70 70 50 Mcl (lOE-9 Nm) 88 36 30 60 Mfs {lOE-7 Nm) 44 18 15 40 u cl ( em/s) 7 5 4 9 ufs (em/s) 230 150 135 325
Tabte 3.1 : Characteristics of 4 mouing-coit meters
The measurement accuraey of G was approx. 3 %. The reproducibilty within one type was found to be of the same order of magnitude. This can fully be subscribed to the above-mentioned accuracy.
A tendency can be seen to the effect that a decrease of the full-scale current results in a decrease of Ucl' although the value of G increases. Hence it is important to use moving-eoil meters which have a low value of Iel'
From the comparison of Ia and II it is found that both types have approximately the same value of G and Iel resulting in approx. the same value of U cl. All types of magnets used seem more or less suitable for making a microanemometer. The type of magnet plays a minor role. Thus in order to be able to establish the usefulness of a moving-eoil meter as microanemometer. other characteristics must play a more important role. such as the diameter of the magnet (determining the diameter of the surrounding sphere) or the maximum length of the pointer.
Summarising, we state tbat the values of Ucl are of the same order
or already lower as those of most commercially available
anemometers. There is good reason to believe tbat the rema.ining step towards our aim of measuring air velocity as low as 1 cmls can be attained with the quantity umi'
As to the values of Ufs we restriet ourselves to the remark tbat all moving-coil meters satisfy one of the demands for measuring air veloeities up to 10 cmls.
3. 5 GENERAL <XINSIDERATIONS
The prototypes described in the previous paragraph seem to be able to measure air veloeities as low as a few cmls. It should be kept in mind tbat, in obtaining these values we based the calculations on the static value of Iel. However, in the event that the feedback system is used, the mimimum detectable air velocity is much lower. resulting in a lower limit of the order of a few mmls. To find the true detection limit of the microanemometer, it bas to be tested in practice, which means that an experimental set-up capable of producing very low air veloeities bas to be built.
As far as the maximum detectable air veloei ty is concerned, the
value obtained of approx. 10 cmls confirms that our
microanemometer can be used to supplement the commercially
available anemometers. Experimental research bas to be done to investigate the exact value of this upper limit by placing it in a wind tunnel.
A
new measuring principle bas been introduced bere, resulting in a new type of microanemometer. It is worth while mentioning tbat in this thesis a restrietion is made as to the type of microanemometer in which the force was measured on the pointer of the moving coil meter itself. However; during recent years much experience bas been gathered with measurements in which a vane was fixed at the far end of the pointer, for instanee see Pluijm et al. '86.In case of a choice between measurements with and without vane, the following remarks should be made:
Where a measurement of a more local air velocity is required, it seems preferabie to use a microanemometer wi th a vane. However. the moment of force exerted on the pointer i tself cannot be neglected. To minimise this effect, one could modify the moving-coil meter and replace the pointer by a beam of much shorter diameter. which is a complicated matter for practical reasons. Another possibility is to construct a microanemometer in which part of the pointer is shielded from the airstrearn, so tbat no force can be exerted on this part of the pointer. Also both rnethods can be combined. Further investigations into this new type of microanemometer will be perforrned in the near future.
4. CALIBRATI«Xf UNIT FOR. Mlar.oANEIDIIITER AT VERY lDI AIR
VELOeiTIES
4. 1 INTRODUCfiON
I f we are to calibrate our microanemometers and establisb tbeir detection limit a calibration unit is needed wbicb can produce air veloeities as low as 1 mm/s, enabling accurate and reproduetbie calibrations wi tb a uniform stationary airflow of known veloei ty to be obtained. This low air veloei ty eaUbration entails two problems:
*No standard facilities are available wbicb can produce air veloeities down to tbe mm/s range.
* Conventional calibration standards generally suffer from insufficient sensitivity (Aydin &Leutbeusser '79}.
Several possible ways to produce low air veloeities are presented in l i terature. The basic principle is always tbe creation of a relative air movement: tbe anemometer is moved tbrough stationary air or set up in an airstream at a fixed place.
* Moving tbe anemometer in stationary air creates practical problems, sucb as tbe influence of mechanica! vibrations (Perry '82). Furtbermore, an absolutely wind-free room is a very hard thing to acbieve as temperature differences cause free convection currents. Bootb & Chong '78 report measurements of a bot-wire anemometer attacbed to a pendulum enclosed in an
insulated box at air veloeities from 3 to 200 cm/s. However, comparison witb a calibration using a low speed wind tunnel sbowed tbe resul ts to be inaccurate owing to tbe airflow around tbe pendulum.
Anotber metbod is to place the anemometer on a sliding cart and pull it througb tbe air, a metbod which is similar to one wbicb is routinely used with 'towing tanks' for ship research (Taneda '56, Ilegbusi & Spalding '83, Tsanis '87). Tabatai et al. '85 used a constant-velocity cart witb a lowest velocity of approx. 2 cm/s.
The movement was effected by connecting tbe cart wi tb ebains or wires to tbe spindie of a manually controlled variabie-speed motor mounted 10 m away from tbe starting position. The time of travel was recorded by clock switcbing at tbe start and end of travel, yielding only tbe average velocity of tbe cart.
*
The second metbod by wbicb to create a relative airflow is to place tbe anemometer in an air stream. Two basic ways can be distinguisbed:1. The metbod using a !ow-speed wind tunnel
The anemometer is placed in tbe working section of a low-speed wind tunnel. A smootb flow is produced tbrougb a nozzle and tbe airflow is tben diverted into tbe working section of tbe tunnel, wbicb is larger in cross-sectional area in order to reduce tbe velocity of tbe air. The pressure drop across tbe smal! nozzle can be related to tbe veloei ty in tbe wind tunnel, knowing the two exit areas and assuming certain veloei ty prof i les across tbe nozzle and working section of tbe tunnel. However, at lower air veloeities tbis conventional technique is inaccurate owing to the instability of jet flows and to the difficulty in obtaining reliable and accurate measurements of tbose pressures corresponding to low velocity flows. Purteil
&
Klebanoff '79 report on a !ow-speed wind tunnel which can create air veloeities as low as 5 cm/s.2. The Piston Flow Metbod
In 1984 an experimental set up based upon tbe Piston Flow Metbod was presented (Pluym et aL '86). Its operating principle was based on the elementary fact that when a closed container of any shape moves at constant velocity, the air inside will follow it at tbe same veloei ty a short time interval af ter tbe start of the movement. It should be noted bere that in 1985 Johannessen, introducing the 'wind wbeel', used a metbod whicb has great simularities witb our Piston Flow Metbod.
4.2 TIIE CALIBRATION UNIT FOR MICROANEMOMETERS
Based upon tbe Piston Flow Metbod and taking into account tbe problems of tbe above-mentioned expertmental set-up, a second.
larger calibration unit was built {Pluijm et al. '86}. A
cylindrical tube (lengtb 600 cm, Dt
=
125 cm} is closed at botb ende witb two circular plates (see fig. 4.1). This tube is placed upon a movable train. By means of a gearbox and a tootbed bar, tbe rotations of a (servocontrolled) motor are transformed into a horizontal displacement of the train. In the laboratory the train can move over about 400 cm. Two safety devices are placed at opposite ends of the stretch (maximum calibration distance}.Fig. 4.1 Schematical representation of the calibration unit for tow air vetoeities
The anemometer is placed ins i de the tube as follows.
At both sides of the laboratory two heavy metal pillars are placed vertically and fixed in the concrete of the floor and the ceiling. A steel cable (diameter 6 mm) is tightened around these pillars. The cable is led into the tube through two small holes (each
8 cm2) in each circular plate. On these cables is placed a wooden standard, upon which the microanemometer is mounted. The position of the standard with the microanemometer inside the cylinder can be changed in the axial direction. By adjusting the pos i tion of the standard itself, it is possible to change the position of the microanemometer in the radial direction.
The position x
1 of the train on the rails is determined by a ten cy
turn potentiometer. Thereto the voltage of the potentiometer
V
t po is registered which is directly related to the pos i ti on of the train. The distance of the microanemometer to the right-hand plate is adjusted befere the measurements, allowing easy calculation of distances to the right and left plate during measurement.The veloei ty U
1 of the train is determined by means of a
cy
tacho-generator. This tacho-generator measures the number of revolutions per second of the motor. yielding a voltage V tacho. Another way to calculate the average velocity is by differentiating x
1.
cy
With this expertmental set-up, veloeities are produced in the mm/s range up to 150 mm/s, divided into three ranges by the gearbox: U
1
<
10 mm/s, 10<
U 1<
70 mm/s and U 1>
70 mm/s. The lowestcy cy cy
velocity met the demands for the microanemometer (see chapter 1) and the highest velocity was determined for practical reasons.
The expertmental set-up can be operated manually as well as automatically by means of a computer. When operated manually the velocity of the train can be changed by means of a potentiometer and the actual velocity of the train, determined from the
4.3 AUfOMATION OF THE EXPERIMENTS
During the development of the calibration unit, the possibility of complete automation was considered. A computer {DEC-LSI} 'and an Eurobussystem, containing an Analog-Digital Converter (A.D.C.} and a Digital-Analog Converter {D.A.C.) is used for the purpose. Both the A.D.C. and the D.A.C. have a computer inaccuracy of 2.5 mV and a dynamic range of lOV. This brought wi th i t an inaccuracy in train position of less than 1 mm. The accuracy in producing and determining the train velocity U
1 is less than 1 %. To minimise
cy
the effect of the A.D.C. inaccuracy on the determination of the air velocity, the output voltage of the feedback system V t
OU
{fig. 4.2) is multiplied in such a way that the maximum inaccuracy due to digitising was 1 %.
The advantages of the automation of the expertmental set up are numerous, for instance:
*
the time necessary for one maasurement is shorter,*
the number of possible measurements per day (or night), is substantially increased, resulting in more reproducible measurements,*
the combination of D.A.C. and A.D.C. yields the possibility of feedback of the velocity and thus control of the constancy of the velocity during the maasurement and*
the computer enables almost any desired time-dependent train velocity, e.g. sinusoirlal fluctuation, to be obtained.In our calibration unit the computer fulfils three basic functions:
*
control of the experiments,*
data acquisition and data processing and*
security functions.Control of the experiments takes place with the aid of D.A.C .. lts output voltage Vda is used for the movement and direction (left/right) of the calibration train. The A.D.C. gives the
possibility of recording. on-line, both the position of the train, as well as the velocity of the train. Combining these two functions, the computer can also be used for security functions: in theevent of an unwanted value.of x 1 the motor is stopped.
cy
In fig. 4.2 a schematic view of the function of the computer is given. Analog Digital Converter A.D.C. p
u
D.A.C. Digi tal Analog Converter V pot V out V tacho p I feedback V daFig. 4.2 Schematic view of the function of the computer
4.4 THE CHARACfERISTIC CALIBRATION MEASUREMENT
4.4.1 Supplementary calibrations
Before the experimental set-up can be used to calibrate anemometers, the calibration unit itself has to be calibrated. This involves determination of the two relations between the D.A.C. voltage Vda and the train velocity Ucyl and the relation between the output voltage of the tacho-generator Vt h and the
. ac o
velocity of the train U
1. This is done in the following way.
First the relationship between Vpot and xcyl is determined. The train is placed at the left and right securi ty device in the laboratory, obtaining resp. vpot,l and vpot,r and the distance between both devices, Llr' measured. At all times xcyl is
4.1
where xcylO is chosen as the middle of the laboratory (Vpot = 0). Several voltages Vda are produced setting the train in motion at a given velocity. Meanwhile the real velocity of the train U
1 is
cy determined by measuring the distance the train moved in 40 (or 20
at higher velocities) seconds. During this measurement Vtacho is also recorded. A characteric result of a supplementary calibration performance is presented in fig. 4.3. where vda vs. ucyl (a} and Vt h vs. U
1 {b) are presented. In each figure three lines are
ac o cy
drawn, corresponding with the three ranges of the gearbox.
5r---.---.---,
T.
vda (V) 0l
vtacho (V) 5 -5~---L---~----~ o~----~---~----~ 0 50 100 150 0 50 100 150 ucyl (mm/s) - ucyl (mm/s)Fig. 4.3 Results of a suppLementary culibration :
From fig. 4.3 it appears that both relations are linear and can therefore be described by:
+ cda· U 1 cy 4.2a
4.2b
where VOda' VOtacho' cda and ctacho are constants, obtained by linear regression.
The supplementary calibrations are performed for train veloeities in both directions. The values of the eight parameters are stored on disc. The reproducibility of these measurements over a number of days proved to be of the order of 2 % and reproducibi l i ty wi thin one day proved to be of the order of 0. 5 %. Th is daily automated calibration does take approx. 20 minutes.
As bas already been mentioned, the calibrations can be automated using the A.D.C. and the D.A.C .. The velocity, the startand the end points of the measurement, as wel! as the number of calibration samples to be taken, can be stipulated beforehand. An
actual calibration measurement proceeds as follows.
Before the measurement the train is sent to the chosen starting position. Generally the starting and end position are chosen symmetrically wi th respect to the pos i ti on of the anemometer in the train. A characteristic course of a measurement is presented in fig. 4.4.
T
u
cylI II III IV V VI VII VIII ---+time
Ftg. 4.4 Characteristic course of a calibration measurement:
U cy. 1 US. time
In this course the following phases can be distinguished:
I train velocity
=
0 ; first determination of the (off-set) output voltage of the microanemometer for 5 seconds.II starting phase: the veloei ty of the train is increased until the destred velocity is reached.
III time interval to cover acceleration effects of the air; no measuring points are taken, in view of computer memory and time.
IV
V
VI VII
CALIBRATION INTERVAL during which 150 or 200 samples of V t' Vt h and V t and other variables are taken. po ac o ou time interval to obtain symmetry with regard to xcylO also no measuring points are taken (see III). ' end phase: the velocity of the train is decreased to:zero. waiting time: 2 minutes are waited to cover
deceleration effects of the air.
VIII train veloei ty
=
0; second determination of the offset output voltage. This second offset voltage is used as a check as to whether the air velocity again equals zero.During maasurement the data are stored on disc, enabling subsequent data processing. During calibration a complete digitised record of the whole calibration run is available for graphical and analytica! preview, so that any anomalous behaviour can immediately be observed and reacted upon.
4. 5 EXPERIMENTS AND DisaiSSION
Several experiments were carried out to establish the usefulness and the limitations of our calibration unit.
*
First the influence of the two holes made in each plate was studied by varying the size between 8 cm2 and 4 cm2. For this purpose Vout was recorded with our microanemometer at air veloeities between 5 and 130 mm/s. From the results we concluded that the size of the holes did not affect the measured Vout· lt should be noted bere that small holes are not always possible, as the size of the holes is directly related to the distance the train can move, in view of the fact that the mass of the anemometer and the standard bend the steel cables.*
During calibration the anemometer is attached to a hollow cylindrical stem resting on a massive waoden standard (see fig. 4.1). The influence of the lengthof the stem on calibration performance was studied. The characteristic height of the stem is of the order of 60 cm, so that the anemometer is located at the centre of the cylinder. Measurements were carried out at heights between 60 and 30 cm and V out was measured wi th the anemometer placed on the standard. The veloeities ranged from 5 to 130 mm/s. The measurements indicated no differences, so that the inf1uence of the standard on the veloei ty profile in the vicini ty of the anemometer can be neglected.*
For many practical purposes it would he of great value if our microanemometer could be calibrated at sinusoidally fluctuating air velocities. Unfortunately our present eaUbration unit does not allow these train velocities. It proved to be possible, however, to produce train veloeities which were a superposition of a constant and a sinusoidally fluctuating velocity {frequencies up to approx. 1 Hz). Calibrations performed at such train veloeities will not be dealt with.5 NUJIERICAL AJW.YSIS OF THE FLOI AROOliD THE III<llOANEIDIETER
5.1 I.NTRODUCfiON
The calibrations required for the development of the microanemometer are performed by means of the eaUbration unit described in chapter 4. Due to the presence of the microanemometer in the calibration unit, the velocity profile is disturbed and the assumption that the air veloei ty at the pos i tion of the pointer equals the veloei ty of the surrounding cylinder might well be inaccurate and deserves special attention.
In li terature, the flow past a sphere bas been studied extensively. The macroscopie hydrodynamica! characteristics, exemplified by the drag coefficient, are well established over a large range of Reynolds numbers by numerous experimental studies. Our interest however is the velocity profile in the vicini ty of the sphere. At very low air velocities, the Stokes approximation is correct and yields an analytica! expression for the flow field. For intermediate Reynolds numbers the complete Navier Stokes equations have to be solved and an analytica! solution is not available owing to the nonlinearity of these equations.
A number of approximate descriptions of the entire flow field through the use of trial stream function polynomials {Chow '79, Kawaguti '55), the boundary-layer assumptions (Schlichting '79) or the finite difference metbod {Jenson '59) are known from literature. These methods, however, are unable to predict accurately the flow behaviour in the region we are interested in
(Hamiliec et al. '67).
We started to calculate the flow in the surroundings of our microanemometer in the eaUbration unit using the Fini te Element Metbod {F.E.M.) .with the Penalty Function Approach (P.F.A.). By calculating the macroscopie hydrodynamica! characteristics, such as the drag coefficient, from the velocity profiles obtainèd with the F.E.M. comparison of these values with those from literature is possible.
Once having calculated the velocity profiles, the influence of the finite dimensions of our calibration unit can be established. In practice this would mean that several cylinders of different diameter have to be used, which would cause a great deal of practical problems. However, by varying the diameter of the calibration unit in our numerical calculations, a measure for the influence of the cylinder wall on the veloei ty profiles can be obtained in a simple way.
5.2 THE NAVIER STOKES EQUATIONS
In order to be able to approach the problem of the flow effects in the vicinity of our microanemometer in the experimental set-up, several assumptions and restrictions have to be made.
*
Only the sphere surrounding the moving-coil meter will be taken into consideration.*
The problem to be solved is restricted to the situation in which the location of the microanemometer in the experimental set-up is on the axis of rotation of the cylinder, so that the problem becomes axisymmetrical.*
The flow is assumed to be stationary.To calculate the velocity profiles in the vicini ty of the we u se cylindrical coordinates (r, '{), z). The
Stokes equations governing the incompressible flow, can be written as:
1 2
(~.v)~
=
-Vp + __ vuRe -s
with the Reynolds number Res =U
•D
/vcyl s flow, dimensionless assuming sphere Na vier steady 5.la 5.lb 5.2
Here u(u ,u ,u ) is the dimensionless velocity, being the actual
- r <p z
air velocity divided by ucyl'
p the dimensionless pressure, obtained by dividing the actual
pressure by 0.5pU cy 21•
ucyl the velocity of the cylinder,
D
5 the diameter of the sphere and
v the kinematic viscosity.
The centre of the sphere is chosen as the origin of the coordinate system and coincides with the axis of symmetry (z-axis}. At the boundaries of the surrounding cylinder the air is assumed to be flowing at velocity ~
=
(0,0,1) parallel to the z-axis.Axial symmetry yields that 8/B<f! is zero and, in addi tion, we assume u
=
0. Hence only half of the ( r, z) plane need be'P
considered. Boundary conditions for ur and uz are dictated by our expertmental set-up, see fig. 5.1 :
*
Çylindrical train (1,2,3} u r=
0*
Sphere surface (5) u r=
0*
Axis of symmetry {4.6) u r=
0 The boundary condition for p reads as*
(3) p=O ' UZ=
1 ,u z=
0 8u /8r = 0 z 5.3a 5.3b 5.3c 5.3dEqns. 5.1, 5.2 and boundary conditions 5.3 together govern the solution of the velocity and pressure profiles.
6 4
1
ê·--·-·--·-·--6-:·--·--·--·--·--·~
ê
2
Fig. 5.1 Geometry and boundary condttions used in the
5.3 THE FINITE ELEMENT METBOD
5.3.1 Broad description of the metbod
As al ready mentioned, the Navier Stokes equations wi th boundary conditions have not yet been solved analytically. Therefore the problem is solved numerically, using tbe Fini te Element Metbod
(F.E.M.) with tbe Penalty Function Approach (P.F.A.).
The F.E.M. is a numerical metbod for solving partial differential equations for a given region (0} and prescribed boundary (r)
conditions. In this particular case tbe equations to he solved are the Navier Stokes equations. The veloei ty and the pressure are wri tten as a linear combination of, in principle. an infini te number of base functions. By restricting tbe number of these functions to a finite number an approximation of the exact salution can be constructed (Chung '78}.
5.3.2 Mathematica! formulation
First tbe region is divided into a finite number of smaller regions called elements (fig. 5.2a) whicb. when joined together, cover the complete region and show no overlap. The result wbich is called tbe 'mesb' is shown in fig. 5.2b,c. In our case the element used is the 7-noded (P2+,Pl) modified triangular Crouzeix-Raviart element (Cuvelier et al.'86).
The next step is to express the unknown veloei ty components and pressure in terms of interpolation functions, called p.(r,z) and
J
op.(r,z) respectively. Polynomials q>. and op. are used such that
J J J
tbey are piecewise continuous on 0 and have a prescribed behaviour for every element (e.g. linear or quadra tic).
and t.(x.)
=
óij are satisfied, where x. isJ -1 -1
nodal point (see fig. 5.2a).
Also q>.(x.) ó
1.J.
J -1
Ftg. 5.2 Velocity: ~ quadratic 7 nodal points ~i Pressure:
+
linear 1 nodal po~nt 0 2 derivativesThe
7-noded(P2+,Pl)
modtfted triangularCrouzetx-Rnvtart eLement (a); enLarged detaiL (b} and complete mesh (c) used for the numertcal anatysts.
The procedure for solving the Navier Stokes equations is discussed briefly. For a detailed description we refer to Cuvelier et al. '86.
In order to linearise the convection term at the left hand side of eqn. 5.la, the Newton Raphson iteration processis used, giving
i i i-1 i i i-1
(~ •v)~
=
(~ •v)~ + (~ ·v)~ (~ i-1 ·v)~ i-1 5.4Substitution of eqn. 5.4 in eqns. 5.1 yields for the i-th iteration: 1 2 i i 1-1 i-1 i i
---- v
u + (~ •v)~ + (~ •v)~ + vpRe
-s 0 5.5bThe principle of the P.F.A. is that the equation of continuity (eqn. 5.lb) is perturbed and replaced by
5.6
where T is a large penalty-function parameter.
Substitution of the interpolation functions ~j(r.z) and ~j(r.z) in the Navier Stokes equations yields a set of 1 inear equations. called the Galerkin equations. In matrix notation these equations can be written as:
