Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

Подождите немного. Документ загружается.

21.7 Laser Doppler Anemometry 705

This eﬀective diameter of the measuring volume has to be smaller than the

diameter of the cross-sectional region of the two laser beams of the incident

optical system:

cos ϕ





cos ϕ

(21.102)

The above equations are equivalent to the requirement that the num-

ber of interference fringes in the intersection area (N

)hastobelarger

than the number of the interference fringes N

which can be seen by the

photodetector. A guideline for this relation is:

≈

(21.103)

With this, one ensures that the outer area of the interference-fringe pattern,

with its bad signal quality, does not have an inﬂuence on the carried out LDA

measurements. A summary of the above equations yields for the number of

fringes within the intersection region of an LDA optical system the following

equation:

≈

∆x

tan ϕ (21.104)

Details of the above considerations can be found in the book by Durst

et al. [21.8]. Multidimensional laser Doppler measurements are possible, but

one pair of beams is required per velocity component, i.e. per measured ﬂow

direction. An optical system which allows one to carry out such measurements

with a two-beam conﬁguration is shown in Fig. 21.44.

In the optical systems represented in Fig. 21.44, the Bragg cells are not in-

cluded. When these are introduced into the two pairs of beams, measurements

with frequency shifts in both pairs of the beams, for two-component measure-

ments, are possible, i.e. the directional recognition of the ﬂow velocities can

be measured at the same time for the two velocity components.

21.7.3 Electronic Systems for Laser Doppler

Measurements

When a scattered particle is traversing the measuring volume of a laser

Doppler optical system, and when the scattered light resulting from this

particle is detected by a photomultiplier, a signal results at the outlet of the

photomultiplier as shown in Fig. 21.45. This signal comprises a low-frequency

component resulting from the Gauss intensity distribution of the laser light.

Furthermore, there is a high-frequency portion that is an inherent part of

the laser Doppler signal. The entire signal thus has a frequency spectrum as

is also shown in Fig. 21.45. It is the high-frequency component of the signal

which is of interest for velocity measurements by LDA.

706 21 Introduction to Fluid-Flow Measurement

Laser

Photo detector 1

Photo detector 2

Collecting lens

Lens

Mask

Prism module 1 Prism module 2

Polarisation filter

Polarisation sensitive

beam splitter

Side view of optical system

Top view of optical system

Fig. 21.44 Two-beam anemometer for two-component LDA measurements of

particle velocity

Frequency s

pectrum

of single signals

Signal produced by a single particle

Properties of doppler signals as obtained by

single particles

Fig. 21.45 Typical Doppler signal of a single scattered particle

In principle, it would be suﬃcient, for determining the Doppler frequency

of a signal originating from a scattered particle, to measure the frequency by

means of a spectrum analyser. The most important components of such an

instrument are indicated in Fig. 21.46. This ﬁgure shows a block diagram of

21.7 Laser Doppler Anemometry 707

Squaring and

smoothing unit

Recorder

Sweep-

generator

I,F,-filter

Mixer

V, C, O,

Spectrum analyser

(1)

(4)

(3)

(2)

( f

)

Fig. 21.46 Working principle of the frequency analyser

the most essential components for scanning the frequency range covered by

the LDA photodetector signal.

In practice, scanning of the expected Doppler frequency range is done

by a ﬁlter whose middle frequency is adjusted to a ﬁxed frequency f

.For

detection of the Doppler frequency, the Doppler signal s

(t)ismixedwith

the signal cos 2πf

t of a voltage-controlled oscillator, and in this way the

Doppler frequency range is scanned by varying the oscillator frequency. When

the signal of the k th particle is represented by:

(t)=a

(t)cos(2πν

t + φ

) (21.105)

for the momentary value of the mixer outlet (an analogue multiplier), the

following signal holds:

(t)=a

(t)cos(2πν

t + φ

)cos2πf

t (21.106)

The amplitude of the mixed signal is proportional to the amplitude of the

photodetector signal and has frequency components f

± ν

Most of the spectrum analysers choose for displacement of the detected

frequency, for their unambiguous operation, the lower of the two frequencies.

When the Doppler frequency to be detected satisﬁes the conditions

−

∆f

≤ f

− ν

≤ f

∆f

(21.107)

a signal passes the ﬁlter and reaches the squaring and integrating part of

the electronics. After squaring and smoothing the frequency-analyser output

signal, the result is recorded.

708 21 Introduction to Fluid-Flow Measurement

The voltage-controlled oscillator is driven by a sawtooth voltage, so that

the frequency of the signal, transmitted by the mixer, increases linearly with

time. As a consequence, the mixture of the Doppler frequencies ν

and os-

cillator frequency, which contribute to the output signal of the analyser, also

increases. When the same sawtooth voltage is used for triggering the x-basis

of a plotter, the Doppler spectrum detected by the frequency analyser can

be plotted. The calibration of the x-axis with respect to frequency is carried

out with the voltage output of a suitable oscillator.

In order to be able also to carry out time-resolved laser Doppler measure-

ments, so-called frequency tracking modulators can be employed. In contrast

to the frequency analyser described above, they permit real-time detection

of the Doppler signal. Frequency tracking demodulation yields an analogue

signal whose voltage is always proportional to the component of the local

ﬂuid velocity which the optical system detects.

Another part of the signal processing with analogous instruments provides

the statistical description of the ﬂow velocity, e.g. via the mean velocity and

the components of the ﬂuctuation velocity, and also quantities which can-

not be obtained by a frequency analysis, such as the turbulence spectrum

and the autocorrelation function of the velocities. Diﬃculties result, how-

ever, from the non-ideal mode of operation of a tracker. They are caused, e.g.

by the often discontinuous signal of the photodetector, which comes from the

fact that only single scattered particles are traversing the measuring volume.

With this demodulated output signal, one does not receive, at every point

in time, information on the momentary ﬂuid motion. Velocity measurements

can only be carried out when a scattering particle is in the measuring vol-

ume. The intermittent measurements can lead to erroneous velocity statistics.

Fluctuations of the recorded Doppler frequencies can also be caused by other

reasons than those of velocity ﬂuctuations in the ﬂuid (e.g. by broadening of

the frequency spectrum due to the presence of the particles in the measuring

volume for ﬁnite time). The statistical evaluation of the measurements, by

the combination of an anemometer and a tracker, is made more diﬃcult.

The essential components of a frequency tracking demodulator are repre-

sented in Fig. 21.47. This ﬁgure shows that the frequency analyzer, discussed

before, and also the tracker, contain three equal components [frequency mixer,

band-pass ﬁlter and voltage-controlled oscillator (VCO)]. Thus the above

explanations, which deal with the properties of the bandpass ﬁlter and the

mode of operation of the mixer, are also of signiﬁcance for the present section

on frequency trackers. The integrator of the tracker corresponds to the time

averaging unit of the frequency analyser. For the discriminator, there is no

comparable component in the frequency analyser. The tracker has, moreover,

in contrast to the frequency analyser, a closed control circuit which drives

the oscillator.

The output signal of the photodetector of an LDA-optical system, similarly

to the processing in a frequency analyser, is mixed with the output signal of

the VCO. Here a signal s

(t), results which is led through a narrow bandpass

21.7 Laser Doppler Anemometry 709

Frequency

discriminator

Integrator

Z,F,-Filter

Mixer

V, C, O,

(1) (2)

(6)

( f

)

Exit

(3)

Fig. 21.47 Functional principle for oﬀset-heterodyne tracker

ﬁlter. Thus, only those signal frequencies of the LDA signal, plus the VCO, are

detected, which are located near the center frequency f

of the ﬁlter. Because

of the small bandwidth of the bandpass ﬁlter (ZF-ﬁlter), the signal-to-noise

ratio (SNR) of the LDA signal to be detected, improves considerably due to

the narrowness of the band pass ﬁlter. A certain improvement of the signal-

to-noise ratio can also be achieved by a ﬁlter in front of the mixer. However,

one has to do this ﬁltering with a ﬁlter bandwidth which is broad in relation

to the Doppler frequency, in order to avoid attenuation of the amplitude of

the Doppler signal in turbulent ﬂows. The frequency discriminator generates

a voltage which triggers the VCO in such a way that the modiﬁcations of the

Doppler frequency by the VCO are compensated. This is explained below.

The integrator controls the transient behavior due to individual LDA signals

and the stability of the control circuit.

Trackers which operate with a narrow bandpass ﬁlter around the cen-

ter frequency f

(see Fig. 21.47) function according to the oﬀset-heterodyne

principle. In laser Doppler anemometry, autodyne trackers have also been

developed, of the kind shown in Fig. 21.48.

Whatever the actual working principles of the LDA frequency tracking

demodulators are, they bring about LDA signals as sketched in Fig. 21.49.

In this ﬁgure the individual Doppler bursts are shown as high-pass ﬁltered

signals, which serve as an input signal into the tracker. Every time the sig-

nal amplitude of an individual LDA signal exceeds a certain threshold value,

the signal is fed to the tracker which then measures the frequency and thus

changes the output voltage of the preceding signal. The output signal of the

tracker therefore has the form of steps, each step being achieved by a new

710 21 Introduction to Fluid-Flow Measurement

cos 2

(

)

0°

90°

V,C,O

Low pass filter

Integrator

sin 2

(

) t

(

)

∝

Fig. 21.48 Functional principle of an autodyne tracker for determining the LDA

signal frequency

Threshold level

High pass filtered signal

Tracker output

Fig. 21.49 Typical input and output signals of trackers for LDA measurements

measurement of the laser Doppler frequency. This tracker output signal, in

the form of steps along the time axis, must not be considered as being dis-

advantageous for measuring the statistically averaged properties of a ﬂow.

The stepwise changes of velocity information take place, with suﬃcient con-

centration of particles, in time intervals which are much smaller than the

characteristic times of the ﬂow, i.e. the time scales at which velocity changes

occur in the ﬂow. It is therefore possible to integrate over several of the

stepwise frequency changes by appropriate electronic devices, to achieve an

averaged output signal for the actual velocity measurements. It is important

to take this pre-averaging into account for precise Doppler measurements.

Finally, a signal-processing system that is extensively employed in laser

Doppler anemometry, the so-called time period measurement system, has to

be explained. Measurement systems of this kind are known as laser Doppler

counters and are extensively applied for LDA measurements. They make use

of signals which, after the photodetector, are fed, often after suitable ampli-

ﬁcation, to a bandpass ﬁlter. It is here, where the actual Doppler frequency

separation from the low-frequency part of the signal takes place. The resultant

signal is sketched at the top of Fig. 21.50. It is processed further in a special

21.7 Laser Doppler Anemometry 711

Doppler signal burst entering

amplitude discriminator

- Pulse train deduced

from upper level crossings

- Pulse train deduced

from zero crossings

- Pulse train deduced

from lower level crossings

- Gate time

- Valid zero crossing

pulse train

Fig. 21.50 Doppler frequency measurement with period measurement system

electronic system, in order to obtain a sequence of pulses which are also il-

lustrated in Fig. 21.50. The entire signal processing of the counter-electronics

functions as follows:

• The symmetrical signal produced by the input stage of the counter-system

is processed by an amplitude discriminator and a zero-position detector

to generate pulse sequences, which are fed to the several logic mod-

ules to check the validity of the frequency information. One possibility

for generating the required pulse sequences that allow precise frequency

measurements, is shown in Fig. 21.50. In practice, this pulse generation

has proven to be a successful method for the production of the precise

information on Doppler frequency.

• The signals of the two-level detectors (Schmitt triggers) and one zero-

position detector are fed to an appropriate logic circuit to generate the

pulse chains 1, 2 and 3. When the Doppler signal crosses the upper trigger

level, it generates a Schmitt trigger output signal, which is used to provide

a switch signal for a logic circuit whose output is then set to one. The

zero-passage detector signal puts this output back to zero. The changing

inﬂuence of the signals from the upper level detector and the zero-passage

detector supplies pulse sequence 1, if the appropriate pulse passes, i.e. the

LDA signal amplitude is satisfactory.

• The pulse sequence 2 is generated by the zero-passage detector only, by

switching the output of an appropriate logic module in such a way that

it switches to and fro between one and zero. A combination of the signals

of the zero-passage detector with those of the Schmitt trigger of the lower

level recognition, yields pulse sequence 3. The output of an appropriate

logic module is set by the lower trigger level and put back by the zero-

passage detector.

• If all three signal chains are present, one will obtain valid information and

use those zero passages for which either output 1 or output 3 is set. In this

way, the inﬂuence of multiple zero passages is suppressed for the largest

part, i.e. good LDA measurements are obtained.

712 21 Introduction to Fluid-Flow Measurement

• The gate for the measurement of the duration of an LDA signal opens

with the ﬁrst valid zero passage and closes one zero passage after the

pulse for which output 1 or output 3 was set. This corresponds to three

zero passages at the end of a Doppler signal, which no longer traverse the

upper or lower trigger level.

The above points show that it is possible, with the aid of counter-systems, to

determine the number of zero passages of an LDA signal, and also the length

of time during which the measured number of pulses is available. With this,

a period-time measurement is possible, which leads to the desired Doppler

frequency of the LDA signals. For each individual LDA signal arriving at

the entrance of the counter-electronics with suﬃciently high amplitude, a

Doppler frequency can therefore be determined. The latter is now processed

further, to obtain mean frequencies and standard deviations.

The measured individual frequencies of laser Doppler signals correspond

to one velocity component, i.e. to U

,t). These individual measurements

have now to be processed further, in order to determine the mean frequency

and the RMS values of the existing deviations from the mean frequency:

>= lim

N→∞



k=1

)

(21.108)

and

<∆f

>= lim

N→∞



[(f

)

− <f

(21.109)

These averaged quantities can be determined easily, as the knowledge of the

individual frequencies, required for averaging, is known from the Doppler

measurements. However, there is a diﬀerence between the time-averaged

quantities which are of importance in turbulence, and the particle-averaged

ﬂow quantities which can be determined from (21.108) and (21.109). This can

be explained in a simple way by means of a sketch of a temporal hypothetical

ﬂow, as shown in Fig. 21.51.

This ﬂow shows a mean motion which is generated by the horizontally

operating piston. Additional ﬂows occur, which once show positively, once

negatively imposed step changes by the motion of the vertically operating

piston. This leads, at the measurement point, to a constant mean velocity

with superimposed step-like ﬂow changes.

Assuming an equal distribution of the scattered particles, it is apparent

that the number of particles which pass the measuring volume depends on

the actual ﬂow velocity, and this can be expressed as follows:

N = c

| U

⊥

|| A

| (21.110)

The number of the measured particles is proportional to the concentration

of the particles in the ﬂuid, and proportional to the ﬂow velocity vertical

to the surface of the control volume, and of course also proportional to the

21.7 Laser Doppler Anemometry 713

•

••

•

••

•

Measuring point

Time averaged

mean velocity

Particle averaged

mean velocity

Fig. 21.51 Dependence of the particle arrival in the measuring volume on the ﬂow

velocity

surface itself. This makes it understandable why in Fig. 21.51 more particles

appear at higher velocities than at lower velocities. This leads to a value of

the particle-averaged velocity that is higher than the ﬂuid time mean velocity,

as indicated in Fig. 21.51. This fact was often called a “biasing error” in Laser

Doppler anemometry and was presented as a principal problem of the LDA

measurement technique. The above explanations make it clear that this only

has to do with the fact that one usually determines ensemble-mean values,

due to their easy determinability from the LDA signals. However, in most

ﬂuid ﬂow studies mean values in terms of time, which are of importance in

turbulence research, need to be measured. This diﬀerence between ensemble

and time averages represents the biasing. This leads to the diﬀerences between

the mean values with respect to time and with respect to particles.

In ﬂuid mechanics, it is usual to determine time averages, e.g. for

determining mean quantities of turbulent ﬂows, which are computed as

follows:

= lim

T →∞

dt and ∆f

= lim

T →∞

∆f

dt, (21.111)

where ∆f

=(f

(t)−f

). The above integration can be carried out digitally

for irregular scanning intervals ∆t

, as shown in Fig. 21.52. This leads to the

following general equations:

For the time average for ∆t

= constant:

= lim

N→∞



k=1

)

∆t



k=1

∆t

with T =



k=1

∆t

(21.112)

714 21 Introduction to Fluid-Flow Measurement

Signal processing with time interval

= constant

Signal processing with time interval

= constant

Fig. 21.52 Determination of the time average of the Doppler frequency for ∆T

constant and ∆t

= constant

= lim

N→∞



k=1

)

with T = N∆t (21.113)

and for the moments which describe the divergence from the mean value the

integration holds that

∆f

= lim

N→∞



k=1

)

−

∆t



k=1

∆t

(21.114)

For a stationary random process, the above equations for mean-value deter-

mination conserve their validity for all scanning intervals ∆t

, provided that

the scanning process and the scanned quantity (Doppler frequency) are not

correlated with one another. The dissolution of a certain frequency in a ﬂow

requires that the ∆t

values are small compared with the time measure of

the ﬂow to be registered.

When the scanning intervals ∆t

are chosen to be constant, i.e. ∆t

∆t = constant, the above equations simplify to

= lim

N→∞



k=1

)

∆t



k=1

∆t

= lim

N→∞



k=1

= <f

> (21.115)