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

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21.7 Laser Doppler Anemometry 695

Observer A

Instantaneous velocity

of particle

}

Observer B

Direction of measured

velocity component

{

}

Scattering particle

Light beam from

laser li

ht source

{

}

Fig. 21.34 Considerations of the Doppler shifted frequency of the scattered light

of the particles. This is the basis of optical velocity measurement by means

of laser Doppler anemometry.

In Fig. 21.34, an arrangement of laser beams is shown, which is suited for

explaining the LDA measurement principle. A beam, emanating from a laser-

light source, has the beam direction {!

} = !

and hits a scattering particle

which scatters light in all directions in space, and consequently also in the

directions which are given by the vectors {k

}= k

and {m

} = m

.The

moving scattered particle has the velocity {U

} = U

and therefore scatters

light having a Doppler shift of the frequency. In the two directions of the

receivers A and B, given by the direction vectors k

and m

, the frequencies

can be derived as

= ν



c − U



and ν

= ν



c − U



(21.87)

Here, the particle generating the scattering beams once acts as a moving re-

ceiver of the arriving laser radiation and at the same time also as a moving

transmitter of the scattered radiation, i.e. the Doppler eﬀect has to be ap-

plied twice, in order to deduce the relationships in (21.87). The simultaneous

detection of the two light signals by observers A and B allows one to detect

the frequency diﬀerence:

∆ν = ν

− ν

− m

) (21.88)

by superposition of the scattered waves. This superposition is carried out

in order to determine the velocity of the scattering particles by a frequency

detectable by available photo detectors.

This superposition of the two waves is carried out since direct measurement

of the frequencies ν

and ν

, which according to (21.87) already show the

desired velocity dependence, is not possible, as the frequencies to be detected

696 21 Introduction to Fluid-Flow Measurement

Lens

Pinhole

Light beam

from laser

Beam Spliter

Interference fringes

measuring volume

Fig. 21.35 Diagram of optical set-up of a two-beam LDA system

are in the range of 10

Hz. Moreover, there is no detection system having the

frequency resolution to measure relative to 10

Hz, the frequency of the laser

radiation, the Doppler shift, which is around 2 ×10

Hz m

−1

s. It is therefore

necessary to determine the velocity-dependent Doppler shift of the laser light

from the frequencies of two scattered light signals that can be superimposed.

This leads to the introduction of the two-beam anemometer, as shown in

Fig. 21.35. A ﬁrst summary of the early work in Laser Doppler anemometry

is given by Buchhave et al. [21.3].

When applying the above derivations to the optical set-up in Fig. 21.35, one

obtains in each direction given by the vector k

the following two frequencies:

= ν



c − U

)

c − U



(21.89)

and

= ν



c − U

)

c − U



(21.90)

From this, the diﬀerence frequency can be computed, for c | U

∆ν =

[(!

)

− (!

)

] (21.91)

This yields for the frequency diﬀerence:

∆ν =

[(!

)

− (!

)

⊥

2sinφ

, (21.92)

where λ is the wavelength of the laser light U

⊥

the velocity component vertical

to the axis of the optical system and ϕ half the angle between the scattering

beams. This relationship shows that the suspended frequency is independent

of the direction of observation. This is a big advantage when employing this

optical system, since a large collecting aperture of the receiving optical system

can be employed to detect the scattered light signal.

21.7 Laser Doppler Anemometry 697

Laser beam 1

{

}

{

}

{

}

Laser beam 2

Fig. 21.36 Visual representation of LDA signals by transparents

In Fig. 21.36, an attempt is made to represent visually, by means of a

method developed by Durst and Stevenson [21.4], the above light superpo-

sition processes which lead to the desired frequency diﬀerence, in order to

measure f

= ∆ν. Figure 21.36 shows the two laser beams of the optical

velocity-measuring instrument and also a moving scattering particle which

was assumed to be present in the measurement volume. This ﬁgure also

shows scattering waves and the lens of the receiving optical system, which is

needed for detecting the scattering light. When superimposing on the scat-

tered wave of the ﬁrst laser beam the second scattered wave of the second laser

beam, then, due to the diﬀerent frequencies of the two scattering beams, the

frequency diﬀerence becomes detectable by means of a photodetector. This

detector yields an electrical signal proportional to the light intensity varia-

tions whose frequency is a measure of the velocity of the particle generating

the two scattered waves. The velocity component vertical to the axis of the

optical system, i.e. perpendicular to the bisector of the two laser beams, is

measured. The measured velocity component is located in the plane set up

by the two laser beams.

The above-explained physical processes of optical velocity measurements

by means of laser beams can also be explained by means of the so-called

interference model; see Fig. 21.37. This model starts from the assumption

that in the intersection volume of the two laser beams, interference fringes

are produced, whose fringe distance is given by the geometry of the optical

set-up and the wavelength of the laser light:

∆x =

2sinϕ

(21.93)

698 21 Introduction to Fluid-Flow Measurement

Interference fringes

in the crossing region

of laser beams

∆

⊥

2 sin

∆

)

The modulation

(

)

depends

on the light intensity differences

of the beams

Fig. 21.37 Visual representation of the interference fringes in the measuring volume

of a laser Doppler anemometer

When a scattered particle is moving through this interference pattern, it

will scatter light whose intensity, for a suﬃciently small particle diameter,

is proportional to the local light intensity in the measuring volume. Because

of this, the intensity of the scattered light shows sinusoidal ﬂuctuations that

are caused by the motion of the particle through the interference pattern.

When the particle has a velocity component U

⊥

perpendicular to the plane

representing the interference fringes, the moving particle needs the following

time to cross a single fringe:

∆t =

∆x

⊥

(21.94)

This brings about a signal frequency which can be expressed as follows:

f =

∆t

⊥

2sinϕ

(21.95)

These considerations lead to the same ﬁnal equation as was derived above

with the help of a Doppler model. In Fig. 21.37, in turn, the modeling of

Durst and Stevenson [21.4] is included, in order to explain the interference

pattern in the measuring volume. Parallel interference fringes are present in

the crossing region of the two incident beams, as explained above, and they

are made visible.

To understand fully the signals occurring in optical velocity measurements,

attention has to be given also to the fact that the incident beams have a

ﬁnite expansion perpendicular to their direction of propagation and that their

intensity distribution over the cross-section shows a Gaussian distribution.

When taking this into account, it is understandable that a particle, that

passes through the center of the measuring volume of an LDA system, brings

about a signal as shown in Fig. 21.37.

When a particle is moving through the measuring volume of an LDA

system, a little away from the center, due to the locally existing intensity

diﬀerences of the two crossing beams, signals are to be expected as indicated

in Fig. 21.38. A signal which originates from a particle that moved through

21.7 Laser Doppler Anemometry 699

Analytical presentation of LDA-Signals

(

)

= Amplitude of

-th signal

= Modulation depth of

-th signal

Fig. 21.38 Analytical representation of LDA signals

Intensity distribulion

of laser light in

measuring volume

Scattering

particle

Intensity distribution as a function of particle size:

Intensity of LDA-signal

Fig. 21.39 Diﬀerent modulation depths with laser Doppler signals

the center of the volume, showing a good signal modulation, serves for com-

parison. Here, the two beams are located in the same plane, and they can

therefore be considered as overlapping well. Away from the center the parti-

cle crosses ﬁrst one beam, then the overlapping area of the two beams and

ﬁnally the area of the second beam. The LDA signal in Fig. 21.39 reﬂects this

ﬁnal form of the signal.

The inﬂuence of the particle size is shown in Fig. 21.39. When particles are

very small, they do not integrate the interference pattern in the measurement

of volume, i.e. the modulation depth of the intensity distribution of the inter-

ference pattern is fully reﬂected in the scattering signal of a particle. In this

700 21 Introduction to Fluid-Flow Measurement

way a signal forms as is shown by the LDA-signal in (A). When the diameter

of a particle is increased, a signal shape develops as sketched in (B). When

the particle corresponds to the size of the distance of the interference fringes,

it is possible that the modulation disappears completely (C). In Fig. 21.39,

equations are given that show these relationships, for a particle assumed to

be square for the considerations carried out here. This form of the particles

is sketched and it is assumed to be moving through the interference pattern.

As far as the determination of the particle velocity direction is concerned,

some additional explanations are necessary. The explanations of optical

velocity measurements given so far, do not permit diﬀerent signals to be gen-

erated for particles with the same velocity, but diﬀerent velocity directions.

Therefore, a particle that moves in one direction through the interference

pattern in Fig. 21.37, will yield the same signal as a particle moving in the

opposite direction, having the same velocity. If, however, one changes the

frequency of a laser beam by a certain amount, this leads to a moving in-

terference pattern in the measuring volume. This can be explained in the

simplest way, when looking at the optical system in Fig. 21.40. It consists of

a diﬀraction grating, which is employed for splitting one laser beam. The two

beams of ﬁrst order are made, with the help of a lens, to cross one another in

the measuring volume. There, in turn, one can imagine the interference-fringe

pattern required for the laser Doppler measurements. This means that the

coherence of the two crossing laser beams is maintained.

The interference fringes forming in the measuring volume can be viewed

as an image of the grid lines present on the diﬀraction grating, and with this

it can be understood that a rotation of the diﬀraction grating brings about

a motion of the interference fringes in the measuring volume. Here, emphasis

has to be laid on the fact that it is not the entire measuring volume that

moves, but only the intensity changes continuously in the measuring volume

with time. This means that only the interference pattern moves and not the

measuring volume. When a particle is now moving in the same direction as

the interference pattern, a smaller frequency is measured than with a motion

directed against the motion of the frequency pattern. When one knows the

Lens

Rotating

grating

Pinhole

Lens

Measuring volume

Photo detector

Pinhole

Fig. 21.40 Diagram of a laser Doppler system based on a rotating diﬀraction grating

21.7 Laser Doppler Anemometry 701

motion velocity of the interference fringes, which is given by the rotation of

the diﬀraction grating, the relative motion can be determined and, hence, also

the sign of the measured velocity component. Simple and easy to understand

introductions into the LDA-measuring technique are provided in references

[2.5] and [2.7].

21.7.2 Optical Systems for Laser Doppler

Measurements

The insights gained in the ﬁrst decade of developments in laser Doppler

anemometry, have led to diﬀerent optical setups for functioning laser Doppler

systems. All of them can be employed for contact-free velocity measure-

ments in ﬂowing ﬂuids. Many available optical systems can be subdivided

into three main groups which nowadays are designated as reference-beam

anemometer, two-beam anemometer and two-scattering-beam anemometer;

see Fig. 21.41. Practical employment of these instruments has shown that the

ﬁrst LDA system shown above, is suitable especially for measurements in

very soiled ﬂuids, where high particle concentrations are present and multi-

ple particles are present in the measuring volume. The second system, the

two-beam anemometer, is particularly suited for measurements, where the

particle concentration is such that, on average, there is less than one particle

in the measuring volume. This can be expected in all liquid ﬂows and gas

ﬂows that appear completely transparent to the human eye.

Fluid flow

Integrated

optical system

Fluid flow

Integrated

optical system

Integrated

optical system

Laser

Lens

Laser

Reference beam anemometer

Two-beam anemometer

Two scattered beam anemometer

Measuring

volume

Collecting lens

Laser light

Mask with

adjustable

slot

Collected light

detected by the

photodetector

Lens

Fig. 21.41 Reference-beam, two-beam and two-scattering-beam optical systems for

laser Doppler measurements

702 21 Introduction to Fluid-Flow Measurement

Beam splitter module

Bragg cell module

Prism module

Laser

Fluid flow

Lens

Collecting lens

Photo detector

Test section

Fig. 21.42 Optical elements of a two-beam laser Doppler anemometer

When a laser beam passes through a ﬂuid, the laser beam generally lights

up intensely and this indicates that there are particles in a range that can-

not be observed by the human eye and which are about a few µm in size.

These particles are suited for velocity measurements with the help of a laser

Doppler system and the two-beam method is particularly suited for reliable

measurements. The two-scattering laser beam method needs to be employed

only in special cases, when measurements of two velocity components have

to be carried out with simple means.

In Fig. 21.42, the essential elements of a two-beam LDA system are shown.

From this diagram, it can be deduced that the optical transmitter system

is essentially composed of the laser light source and a beam-splitter unit,

followed by a lens. A frequency-shifting unit, consisting of Bragg cells, is also

included for measuring the direction of the ﬂow. For the collection of the light,

another lens and a photomultiplier, with an appropriate pinhole, are required.

In the intersection region of the two beams of this optical set-up, one can

imagine that the interference fringes, explained in Sect. 21.7.1, form for the

actual measurements. To obtain good LDA signals, it is essential that these

interference fringes are fully modulated, i.e. that intensity becomes zero in the

dark part of the interference fringe pattern. This can be achieved by matching

the intensities of the two beams, and in addition by employing optimal optical

systems, i.e. optical systems with the same optical pathlengths of the two

beams. This latter requirement yields in the measuring volume to zero phase

diﬀerences of the superimposed laser-light waves.

For precise laser Doppler measurements, it is necessary to choose the cor-

rect laser. As most lasers have an outlet showing several axial modes, it

is necessary to keep the optical pathlengths of the two laser beams of the

system to be almost the same. This is guaranteed by employing an optical

beam-splitting prism, as indicated in Fig. 21.42. This prism has further ad-

vantages that recommend it for employment for practical measurements. It

is insensitive to adjustment and, hence, rotation of the prism in the plane,

as shown in Fig. 21.42, does not lead to a directional change of the parallel

beams leaving the prism. These beams are always parallel to the incident

beam and always have the same distance in relation to the optical axis of the

21.7 Laser Doppler Anemometry 703

(

)

(

)

min

Fig. 21.43 Sketch of laser beams to explain the Gaussian beam behavior in optical

systems

system. By choosing the prism shown in Fig. 21.42, LDA instruments become

insensitive to adjustment and are suited for practical ﬂow investigations.

Laser beams show a behavior that is usually referred to as that of a Gaus-

sian beam (Fig. 21.43). For f →∞and ρ

→∞, one can derive from

relationships, as indicated by Durst and Stevenson [21.4], analytical expres-

sions for the variation of the wave-front curvature R(z) and the beam radius

s(z):

R (z)=z



πω

λz



and s (z)=ω



λz

πω



(21.96)

With the help of these equations, the radius of the two beams and the wave-

front curvature at the lens of an LDA system can be computed:

= ω



λz

πω



(21.97)

and

= δ



πω

λσ



(21.98)

The expression for the light intensity is

(r)=



πω

fλ



exp



−



(21.99)

704 21 Introduction to Fluid-Flow Measurement

The inﬂuence of the properties of focused Gaussian beams on the mode

of operation of laser Doppler anemometers was investigated by Durst and

Stevenson [21.4] and Hanson [21.2]. When using a single lens to focus the

two incident beams, as is usually the case in two-beam systems, the axes of

the beams cross one another in the focal region of the lens. When the beam

waists are not adjusted to this region, this has an increased measuring volume

as a consequence. In addition, it could be shown by Durst and Stevenson [21.4]

that the curvature of the wave fronts, which occurs inside the crossing region

of the two beams, can lead to signiﬁcant changes of the Doppler frequency,

when the particles traverse diﬀerent parts of the measuring volume. If prop-

erly set up LDA systems are employed, the mentioned eﬀects are often very

small and they were usually ignored in ﬂuid ﬂow measurements with laser

Doppler anemometers, either because they were not known, or because the

advantages of a set-up with a single lens outweighed these small discrepan-

cies. For some applications, e.g. in laser Doppler measurements over large

distances, investigations in extended ﬂow ﬁelds, etc., the indicated errors can

be of signiﬁcance, so that appropriate steps have to be taken to ensure an

optimal beam intersection region. This can be achieved in two ways:

• The waists of the two laser beams are laid into the back focal region of

the transmitter lens of the LDA optical system.

• An additional optical component, consisting of a convex and a concave

lens, is put between the laser and the lens of the LDA optical system. This

system is used for choosing freely the position of the beam contraction

with regard to the main lens of the optical system.

The above means that, to carry out now reliable LDA measurements, one

has to ensure that the beams in the measuring volume of a laser Doppler

anemometer are focused in such a way that we obtain parallel interference

fringes, with a high modulation depth, i.e. the minimum of the light intensity

should almost be zero. As far as the optical set-up of an LDA system is

concerned, the following should be mentioned:

• The lens in front of the photodetector has to be chosen such that the

equation:

Mλ

2sinϕ

(21.100)

holds, where d

is the number of interference fringes which the photo

multiplier sees, M is the optical enlarging relation of the detection system

(M = b/a), λ the wavelength of the laser light and ϕ is half the angle

between the two incident laser beams.

The diameter of the eﬀective measuring volume can then be calculated as:

= ∆xN

λN

2sinϕ

(21.101)