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Laser Doppler Anemometry

The basic principles of laser Doppler anemometry (LDA) are summarized here. For a more comprehensive treatment of these principles, the reader is referred to standard text books such as those by Durst et al. Durst76, Durrani and Greated Durrani77, Somerscales Marton81, Absil Absil95, and Albrecht et al. Albrecht03 Aspects of LDA that are typical for a three-component system are described in some detail in Section 1.3.

Introduction

In LDA the Doppler-shift is determined of light scattered by a small particle that moves with the flow. This Doppler-shift provides a measure for the velocity of the particle, and, therefore, for the flow velocity. Since the introduction of the technique by Yeh and Cummins Yeh the use of LDA has become widespread in both research and industrial applications. The main advantage of the technique over conventional measuring techniques, such as hot-wire anemometry (HWA) and pressure probes, is that it does not require a physical probe in the flow, i.e. it is a non-intrusive technique. Therefore, the flow is not disturbed during a measurement. Other advantages of the technique are:

The combination of the different items makes the technique ideally suited for measurements in turbulent flows. Both the magnitude and the direction of the instantaneous velocity vector can be measured. It does not only give accurate information on relatively simple statistics, such as mean velocities and Reynolds stresses, but the non-intrusive nature also enables the measurement of more complex quantities such as spatial correlation functions.

The list of advantages is impressive, it resembles the specification of an ideal measuring instrument. However, it is not difficult to list some serious drawbacks of LDA:

However, the main disadvantage lies in the complexity of the measuring technique. The complexity not only results in relatively high cost of purchasing an LDA system, it also requires an experienced operator who is familiar with all the peculiarities of the measuring technique.

Basic Principles of LDA

Doppler frequency

The Doppler effect forms the basis of LDA. Light scattered by a small moving particle undergoes a shift in frequency. This frequency shift is called the Doppler frequency and it is related to the velocity of the particle. Below, the relationship between the Doppler frequency and the velocity is derived.
Figure 1.1 : The light-scattering configuration.

Consider Fig. 1.1 which shows a light source that generates a plane light wave with frequency f0. The direction of propagation of the plane wave is given by the unit vector $ \vec{{e}}_{i}^{}$. In complex notation the plane wave is given by

Ei($\displaystyle \vec{{x}}\,$) = Ei0e-i(2$\scriptstyle \pi$f0t-k0$\scriptstyle \vec{{e}}_{i}$$\scriptstyle \vec{{x}}\,$+$\scriptstyle \phi_{i}$), (1.1)
where $ \phi_{i}^{}$ is the initial phase and k0 = 2$ \pi$/$ \lambda_{0}^{}$ is the wave number. The wavelength $ \lambda_{0}^{}$ is related to the frequency of the light source as $ \lambda_{0}^{}$ = c/f0 where c is the speed of light. A particle passing through the plane wave scatters light in all directions and some of the light will be received by a detector. The orientation of the detector is determined by the unit vector $ \vec{{e}}_{d}^{}$, and the detector is located at $ \vec{{r}}\,$ = $ \vec{{r}}_{d}^{}$. The spherical wave emitted by the particle may be represented by [#!Adrian83!#]:

Es($\displaystyle \vec{{r}}\,$ - $\displaystyle \vec{{x}}\,$) = $\displaystyle {\frac{{E_{s_0} \sigma_s}}{{ \left\vert \vec{r}-\vec{x} \right\vert}}}$ e-i(2$\scriptstyle \pi$fst-ks$\scriptstyle \left\vert\vphantom{ \vec{r}-\vec{x} }\right.$$\scriptstyle \vec{{r}}\,$ - $\scriptstyle \vec{{x}}\,$$\scriptstyle \left.\vphantom{ \vec{r}-\vec{x} }\right\vert$+$\scriptstyle \phi_{s}$), (1.2)
where $ \sigma_{s}^{}$ depends on the scattering characteristics of the particle. At the particle ($ \vec{{r}}\,$ = $ \vec{{x}}\,$) it follows from Eq (1.1) and Eq (1.2) that

2$\displaystyle \pi$f0t - k0$\displaystyle \vec{{e}}_{i}^{}$$\displaystyle \vec{{x}}\,$ + $\displaystyle \phi_{i}^{}$ = 2$\displaystyle \pi$fst + $\displaystyle \phi_{s}^{}$. (1.3)
This yields the following expression for the phase $ \Phi$ of the spherical wave at the detector ($ \vec{{r}}\,$ = $ \vec{{r}}_{d}^{}$)
$\displaystyle \Phi$ = 2$\displaystyle \pi$f0t + $\displaystyle \phi_{i}^{}$ - k0$\displaystyle \vec{{e}}_{i}^{}$$\displaystyle \vec{{x}}\,$ - ks$\displaystyle \left\vert\vphantom{ \vec{r}_d - \vec{x} }\right.$$\displaystyle \vec{{r}}_{d}^{}$ - $\displaystyle \vec{{x}}\,$$\displaystyle \left.\vphantom{ \vec{r}_d - \vec{x} }\right\vert$ (1.4)
  = 2$\displaystyle \pi$f0t + $\displaystyle \phi_{i}^{}$ - k0$\displaystyle \vec{{e}}_{i}^{}$$\displaystyle \vec{{x}}\,$ - ks$\displaystyle \left(\vphantom{ R - \vec{e}_d \frac{\textup{d} \vec{x} }{\textup{d} t } t }\right.$R - $\displaystyle \vec{{e}}_{d}^{}$$\displaystyle {\frac{{\textup{d} \vec{x} }}{{\textup{d} t }}}$t$\displaystyle \left.\vphantom{ R - \vec{e}_d \frac{\textup{d} \vec{x} }{\textup{d} t } t }\right)$ ,  

where it is assumed that at time t = 0 the distance between the particle and the detector is R. Furthermore, d$ \vec{{x}}\,$/dt $ \equiv$ $ \vec{{v}}\,$ is the velocity vector of the particle at t = 0. The frequency of the scattered light as seen by the detector, fw, is proportional to the time derivative of $ \Phi$, i.e.

2$\displaystyle \pi$fw = $\displaystyle {\frac{{\textup{d} \Phi}}{{\textup{d} t}}}$ = 2$\displaystyle \pi$f0 + $\displaystyle \vec{{v}}\,$(ks$\displaystyle \vec{{e}}_{d}^{}$ - k0$\displaystyle \vec{{e}}_{i}^{}$) . (1.5)
If the velocity of the particle is small compared to the speed of light, it can be assumed that k0 $ \approx$ ks in Eq (1.5), so that the frequency of the scattered light at the detector, fw, becomes

fw = f0 + $\displaystyle {\frac{{\vec{v} (\vec{e}_d - \vec{e}_i)}}{{\lambda_0}}}$. (1.6)
The second term on the right-hand side of Eq (1.6) is known as the Doppler frequency. It contains information on the component of the velocity in the direction of the vector $ \vec{{e}}_{d}^{}$ - $ \vec{{e}}_{i}^{}$. This vector is determined by the geometry of the optical arrangement of the LDA.


Heterodyne detection

It is the task of the detector to generate an output signal from which the Doppler frequency fD can be determined. The direct measurement of the Doppler frequency requires a very high resolution of the detector, because the Doppler frequency is much smaller than the frequency of light; typically fD/f0 $ \approx$ 10-13. In the low-velocity range, say $ \left\vert\vphantom{ \vec{v} }\right.$$ \vec{{v}}\,$$ \left.\vphantom{ \vec{v} }\right\vert$ < 300 m/s, the Doppler frequency can be determined with an ``optical mixing'' or ``heterodyne'' technique in conjunction with a square-law detector. The essence of heterodyning is that when two light waves with slightly different frequencies, fw1 and fw2, are mixed on the surface of a square-law detector, the output signal oscillates with the difference frequency fw1 - fw2.

Figure 1.2: The optical arrangement for the dual-beam heterodyne LDA.
\begin{figure}\centerline{ \epsfxsize=79mm \epsffile{plaatjes/dop2.prn} }\end{figure}

Because of its superior signal-to-noise ratio, the most widely used optical configuration is the dual-beam heterodyne configuration. Figure 1.2 shows the optical arrangement for the dual-beam configuration. Eq (1.6) can be applied to both incident beams, resulting in

$\displaystyle \left.\vphantom{ \begin{array}{ll} f_{w_1} = f_0 + \vec{v}(\vec{... ..._2} = f_0 + \vec{v}(\vec{e}_d-\vec{e}_{i_2}) / \lambda_0 & \end{array}}\right.$$\displaystyle \begin{array}{ll} f_{w_1} = f_0 + \vec{v}(\vec{e}_d-\vec{e}_{i_1... ... f_{w_2} = f_0 + \vec{v}(\vec{e}_d-\vec{e}_{i_2}) / \lambda_0 & \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{ll} f_{w_1} = f_0 + \vec{v}(\vec{... ...2} = f_0 + \vec{v}(\vec{e}_d-\vec{e}_{i_2}) / \lambda_0 & \end{array}}\right\}$  $\displaystyle \Rightarrow$  fD $\displaystyle \equiv$ fw2 - fw1 = $\displaystyle {\frac{{\vec{v}(\vec{e}_{i_1}-\vec{e}_{i_2})}}{{\lambda_0}}}$ . (1.7)
The unit vectors $ \vec{{e}}_{{i_1}}^{}$ and $ \vec{{e}}_{{i_2}}^{}$ indicate the direction of the incident beams, and the Doppler frequency fD is now defined as the difference between fw1 and fw2. Inspection of Eq (1.7) shows that fD is a measure for the velocity component in the direction of $ \vec{{e}}_{{i_1}}^{}$ - $ \vec{{e}}_{{i_2}}^{}$. The Doppler frequency is independent of the orientation of the detector with respect to the incident beams, which enables the use of large apertures to collect more scattered light.

When a particle passes through the overlap region of the incident beams, i.e. the so-called measuring volume, it scatters light in all directions. A square-law detector, usually a photomultiplier tube, then receives light with two slightly different frequencies, fw1 and fw2. The relationship between the output signal y(t) and the input signal x(t) of a photomultiplier is given by [#!Marton81!#]

y(t) = $\displaystyle {\frac{{S}}{{T_s}}}$$\displaystyle \int_{{t-T_s/2}}^{{t+T_s/2}}$x(t')2dt', (1.8)
where Ts is the time constant of the photomultiplier and the constant S represents the radiant sensitivity. According to Eq (1.5) the input signal can be written as

x(t) = a1cos(2$\displaystyle \pi$fw1t + $\displaystyle \phi_{1}^{}$) + a2cos(2$\displaystyle \pi$fw2t + $\displaystyle \phi_{2}^{}$) , (1.9)
where $ \phi_{1}^{}$ - $ \phi_{2}^{}$ is the phase difference between the two light waves. The phase difference is assumed to be constant, which indicates the need for a coherent light source such as a laser. On combining Eqs (1.7) through (1.9) the following expression for the output signal of the photomultiplier is obtained (assuming Tsfw1 $ \gg$ 1 and Tsfw2 $ \gg$ 1)

y(t) = $\displaystyle {\frac{{1}}{{2}}}$Sa12a22 + Sa1a2$\displaystyle {\frac{{ \vert\sin( \pi f_D T_s)\vert }}{{ \pi f_D T_s}}}$cos(2$\displaystyle \pi$fDt + $\displaystyle \phi_{1}^{}$ - $\displaystyle \phi_{2}^{}$). (1.10)
For small values of the time constant Ts, say Ts $ \approx$ 10-9 s, Eq (1.10) reduces to the following well-known expression for the output signal of the photomultiplier

y(t) = $\displaystyle {\frac{{1}}{{2}}}$Sa12a22 + Sa1a2cos(2$\displaystyle \pi$fDt + $\displaystyle \phi_{1}^{}$ - $\displaystyle \phi_{2}^{}$). (1.11)
The first term on the equation's right-hand side is known as the pedestal; it is the result of the spatial distribution of the light intensity in the overlap region of both beams. The second term, called the Doppler burst, carries the desired information, because it oscillates with the Doppler frequency fD.

Directional ambiguity

Referring to Fig. 1.2, it is easy to see that the expression for the Doppler frequency, Eq (1.7), can be rewritten as

fD = $\displaystyle {\frac{{\vec{v}(\vec{e}_{i_1}-\vec{e}_{i_2})}}{{\lambda_0}}}$ = $\displaystyle {\frac{{2 \sin(\theta/2)}}{{\lambda_0}}}$ |$\displaystyle \vec{{v}}\,$| sin$\displaystyle \alpha$, (1.12)
where $ \theta$ is the angle between the unit vectors $ \vec{{e}}_{{i_1}}^{}$ and $ \vec{{e}}_{{i_2}}^{}$, i.e. $ \theta$ is the crossing angle of the incident laser beams. Furthermore, $ \alpha$ is the angle between the velocity vector and the x-axis. The frequency difference fD can be positive or negative depending on the value of $ \alpha$. However, the output of the photomultiplier cannot distinguish between positive and negative values of fD because cos(- fD) = cos(fD).
Figure 1.3 : The effect of frequency shift on the relationship between the particle velocity and the frequency of the photomultiplier output signal.
\begin{figure}\centerline{ \epsfxsize=58mm \epsffile{plaatjes/shift.prn} } \par \end{figure}
Figure 1.4 : The interference of two plane light waves.
\begin{figure}\centerline{ \epsfxsize=64mm \epsffile[142 204 714 516]{plaatjes/fmodel.prn} } \end{figure}
As a result, the LDA in its basic form is unable to determine the sign of the velocity. The insensitivity to the direction of the particle velocity is usually referred to as the ``directional ambiguity.'' The common method to remove this ambiguity is frequency shifting. In that case the frequency of one of the incident beams in Fig. 1.2 is shifted by a constant value fs. This can be achieved with an acousto-optic Bragg cell. Due to the frequency shift the relationship between Doppler frequency and particle velocity becomes (assuming fs $ \ll$ f0)

fD = fs + $\displaystyle {\frac{{2 \sin(\theta/2)}}{{\lambda_0}}}$ |$\displaystyle \vec{{v}}\,$| sin$\displaystyle \alpha$ , (1.13)
as illustrated in Fig. 1.3. If the shift frequency fs is chosen larger than the Doppler frequency that corresponds to the smallest anticipated velocity in the flow, vmin, each value of | fD| is uniquely related to one velocity value, and, as a consequence, the directional ambiguity is removed. In practice, one usually sets the shift frequency fs about two times larger than the Doppler frequency that corresponds to vmin [#!Tropea86!#].

Fringe model

An alternative procedure to derive the relationship between the Doppler frequency and the velocity for the dual-beam LDA, Eq (1.12), is given by the ``fringe model,'' which is due to Rudd Rudd69. The fringe model is often used to visualize different aspects of the dual-beam configuration, such as the nature of the detector output signals in case the incident beams are improperly aligned [#!Durst79!#]. It also gives an interpretation of the proportionality constant between the Doppler frequency and the velocity in Eq (1.12). However, the fringe model should be considered with some reserve, because it is incorrect in the sense that it ignores the fact that heterodyning takes place on the surface of the photomultiplier and not at the particle, see Durst Durst82. However, most of the predictions of the fringe model are in accordance with the Doppler theory, as will be shown below.

If the incident beams shown in Fig. 1.4 are properly aligned, their wavefronts are nearly plane in the overlap region, so that the light waves can be described with Eq (1.1). The intensity of the light in the overlap region of both beams is then given by

I = (E10 + E20)(E10* + E20*) (1.14)
  = E102 + E202 +2E10E20cos(2k0y sin($\displaystyle \theta$/2) + $\displaystyle \phi_{1}^{}$ - $\displaystyle \phi_{2}^{}$) ,  

where $ \theta$ is the angle between the unit vectors $ \vec{{e}}_{{i_1}}^{}$ and $ \vec{{e}}_{{i_2}}^{}$, y is a coordinate in the direction of $ \vec{{e}}_{{i_1}}^{}$ - $ \vec{{e}}_{{i_2}}^{}$ and $ \phi_{1}^{}$ - $ \phi_{2}^{}$ is the phase difference between the two light waves. According to Eq (3.14) the intensity varies periodically in y, and the distance between two consecutive lines of constant intensity in the interference pattern is given by

df = $\displaystyle {\frac{{\lambda_0}}{{2 \sin(\theta/2)}}}$ . (1.15)
The quantity df is known as the fringe distance, and inspection of Eq (1.12) reveals that it is the inverse of the proportionality constant between the Doppler frequency and the velocity. A small particle passing through the interference pattern with a velocity component in the y-direction of v(= $ \left\vert\vphantom{ \vec{v} }\right.$$ \vec{{v}}\,$$ \left.\vphantom{ \vec{v} }\right\vert$sin$ \alpha$), scatters light with an intensity that is proportional to the local value of I. The intensity of the scattered light then oscillates with frequency

$\displaystyle {\frac{{v}}{{d_f}}}$ = $\displaystyle {\frac{{2 \sin(\theta/2)}}{{\lambda_0}}}$$\displaystyle \left\vert\vphantom{ \vec{v} }\right.$$\displaystyle \vec{{v}}\,$$\displaystyle \left.\vphantom{ \vec{v} }\right\vert$sin$\displaystyle \alpha$. (1.16)
It follows from a comparison with Eq (1.12) that this is identical to the Doppler frequency.

The fringe model can also be used to visualize the effects of applying a frequency shift to remove the directional ambiguity. If in Fig. 1.4 the frequency of one of the beams, say beam 1, is increased with a value fs, the intensity of the light in the overlap region becomes

I = E102 + E202 +2E10E20cos(2$\displaystyle \pi$fst + 2k0y sin($\displaystyle \theta$/2) + $\displaystyle \phi_{1}^{}$ - $\displaystyle \phi_{2}^{}$) . (1.17)
The fringes in the interference pattern now move with velocity vs = dffs in the positive y-direction. As a result, a detector sees intensity variations with a frequency

fs + $\displaystyle {\frac{{v}}{{d_f}}}$ = fs + $\displaystyle {\frac{{2 \sin(\theta/2)}}{{\lambda_0}}}$$\displaystyle \left\vert\vphantom{ \vec{v} }\right.$$\displaystyle \vec{{v}}\,$$\displaystyle \left.\vphantom{ \vec{v} }\right\vert$sin$\displaystyle \alpha$ = fD , (1.18)
which is identical to Eq (1.13), the result obtained using the Doppler theory.


Amplitude bias

Durao and Whitelaw DurWhi79 have shown experimentally that there is a relationship between the amplitude of a Doppler burst and the particle velocity. There study revealed that there is a tendency for low-speed particles to produce high-amplitude Doppler bursts, and vice versa. Through this mechanism low-velocity particles have (on the average) a larger probability of being detected and validated by the LDA signal processor than high-velocity particles. This bias towards low velocities was termed ``amplitude bias'' by Durao and Whitelaw. They argued that the amplitude bias was due to the fact that fast moving particles (on the average) spend less time in the measuring volume than slow particles. The fast moving particles scatter less photons and, therefore, produce Doppler bursts with smaller amplitude.

It was already shown in Section 1.2.2 (Eq (1.10)) that, in principle at least, the amplitude A of a Doppler burst depends on the Doppler frequency through the term

A = Sa1a2$\displaystyle {\frac{{ \vert\sin( \pi f_D T_s)\vert }}{{ \pi f_d T_s}}}$ , (1.19)
where Ts is the time constant of the photomultiplier and fD is the Doppler frequency. Substitution of Eq (1.18) in Eq (1.19) yields the following expression for A

A = Sa1a2$\displaystyle {\frac{{ \vert\sin( \pi (f_s + \frac{v}{d_f} ) T_s)\vert }}{{ \pi (f_s + \frac{v}{d_f} ) T_s}}}$ , (1.20)
where fs is the shift frequency, v is the velocity component and df is the fringe distance. Figure 1.5 depicts the amplitude A as a function of v for several values of the photomultiplier time constant Ts ranging between 2 ns and 32 ns. Furthermore, it is assumed that df = 3 $ \mu$m and fs = 40 MHz.
Figure 1.5 : The amplitude A versus the velocity v for several values of Ts ranging between 2 ns and 32 ns.
\begin{figure}\centerline{ \epsfxsize=68mm \epsffile[62 74 488 473]{plaatjes/ambias.eps} } \end{figure}
Figure 1.5 shows that the amplitude A will significantly vary with the particle velocity only when the value of Ts is large. Clearly, photomultipliers with large time constants are not suited for LDA, because such photomultipliers give rise to the amplitude bias. Figure 1.5 also shows that the amplitude A is practically constant for small values of Ts. This means that the dependence of the amplitude A on the particle velocity can be conveniently ignored when a so-called ``fast response'' photomultiplier is used, which is usually the case.

Particle characteristics

The light-scattering particles form an essential element of the LDA measuring system. In each application the suitability of the particles must be determined in the same way as any other element of the LDA instrumentation. In general, it is highly appreciated if the particles are cheap, easy to generate, non-corrosive and non-toxic. However, the suitability of the particles for application in LDA mainly depends on their dynamical and optical characteristics.

The optical characteristics should be such that the particles scatter light with sufficient intensity for the photodetector to generate high-quality Doppler signals. Investigations based on Mie's scattering theory, e.g. Durst Durst82, show that the amplitude and the visibility of the Doppler signals are dependent on the particle diameter, the refractive index of the particle material, the wavelength of the laser light, the angle between the incident laser beams and the aperture and orientation of the receiving optics. Generally speaking, the amplitude and visibility of the Doppler signals increase with increasing particle size and increasing index of refraction.

The dynamical characteristics of particles determine their ability to accurately follow the fluctuations in the fluid velocity even at high frequencies. The motion of a rigid, spherical particle in a viscous flow is governed by the Basset-Boussinesq-Oseen (BBO) equation, see Somerscales Marton81. Solutions of the BBO equation are discussed by Hjelmfelt and Mockros Hjelm66. A simplified equation of motion is given by (see Somerscales Marton81)

$\displaystyle \rho_{p}^{}$$\displaystyle {\frac{{\pi d_p^3}}{{6}}}$$\displaystyle {\frac{{\textup{d} u_p}}{{\textup{d} t}}}$ = 3$\displaystyle \pi$$\displaystyle \nu$$\displaystyle \rho_{f}^{}$dp(uf - up), (1.21)
where $ \nu$ is the fluid kinematic viscosity, dp is the particle diameter, up and uf are the particle and fluid velocities and $ \rho_{p}^{}$ and $ \rho_{f}^{}$ are the particle and fluid densities, respectively. The equation's left-hand side represents the force to accelerate the particle. The term on the right-hand side is the drag of the particle for which Stokes' drag law is used. The validity of Eq (1.21) is restricted to large values of the density ratio $ \sigma_{r}^{}$ $ \equiv$ $ \rho_{p}^{}$/$ \rho_{f}^{}$ and not too large acceleration. Furthermore, the effects of, for example, centrifugal forces, electrostatic forces and gravity are ignored.

Eq (1.21) will be used here to formulate a criterion for the diameter of the particles. Following Hjelmfelt and Mockros Hjelm66 the fluid velocity and the particle velocity are expressed in terms of Fourier components. Substitution of uf = ei$\scriptstyle \omega$t and up = $ \eta$($ \omega$)ei$\scriptstyle \omega$t in Eq (1.21) yields the amplitude ratio |$ \eta$| as

|$\displaystyle \eta$($\displaystyle \omega$)| = $\displaystyle {\frac{{\Omega}}{{\sqrt{\Omega^2 + \omega^2 }}}}$        with        $\displaystyle \Omega$ = $\displaystyle {\frac{{18 \nu}}{{\sigma_r d_p^2}}}$ . (1.22)
The amplitude ratio can be interpreted as a measure for the sensitivity of the particles to changes in the fluid velocity. It is seen from Eq (1.22) that the particle motion is attenuated at high frequencies. The maximum diameter of a particle that follows the velocity fluctuations up to 1 kHz, 5 kHz and 10 kHz for |$ \eta$| = 0.99 can be determined from Eq (1.22). For a number of frequently used seed materials the thus obtained diameters are listed in Table 1.1. From these results it can be concluded that oil particles with a diameter of typically 1 $ \mu$m accurately track the velocity fluctuations in low-speed air flows. High-speed flows generally require smaller particles, because of the energy of the velocity fluctuations at higher frequencies.

Table 1.1 : Maximum particle diameter for various seed materials.
seed density ratio $ \sigma_{r}^{}$ maximum diameter $ \left[\vphantom{ \mu \textup{m} }\right.$$ \mu$m$ \left.\vphantom{ \mu \textup{m} }\right]$

material

 (in air) 1 kHz 5 kHz 10 kHz
silicone oil 620 3.1 1.4 1.0
rizella oil 711 2.9 1.3 0.9
polystyreen 865 2.6 1.2 0.8
teflon 1800 1.8 0.8 0.6
titanium oxide 3500 1.3 0.6 0.4


The presence of a shock wave in a supersonic flow provides a further motivation to use submicron particles. Due to the strong decelerations across the shock wave the particle velocity lags the fluid velocity. This phenomenon has been studied by Yanta et al. Yanta71 and more recently by Maurice Maurice92. The particle lag may result in a severe overestimation of the mean velocity and the turbulence intensity at locations directly downstream of the shock wave if too large particles are used. In general, particles that accurately follow the abrupt velocity changes in supersonic flows should have diameters less than 0.3 $ \mu$m.

There are flows with practical relevance for which Eq (1.21) is invalid. In vortical flows the centrifugal forces induce a migration of particles away from the core region (for $ \sigma_{r}^{}$ > 1), thus reducing the particle concentration in the core. The particle concentration may become so low that LDA measurements in the core become almost impossible as reported by Meyers and Hepner MeyHep88. Details on the particle motion in vortical flows can be found in e.g. Dring and Sou Dring78.


Signal processor

The principle task of a signal processor is to extract the Doppler frequency (i.e. velocity) from the photomultiplier output signal. Usually, the signal processor also measures other quantities such as the arrival time of the particles and the duration of the Doppler bursts, i.e. the transit time of the particles. Two commonly used signal processors for sparsely seeded flows are ``counter processors'' and ``spectrum analyzers.'' A detailed discussion of the characteristics of the two types of processors and a comparison between their performances is beyond the scope of this thesis. Instead, this section describes only the basic principles of one representative of the latter type of signal processor. The processor to be described is the Burst Spectrum Analyzer (BSA) which is manufactured by Dantec, and available since the late 1980s.

The BSA processor performs a spectral analysis of the bandpass-filtered output signal of the photomultiplier. The Doppler frequency then follows from the location of the peak in the computed power spectrum. The basic principles of the Dantec BSA are illustrated in Fig. 1.6.

Figure 1.6 : The basic principles of the Dantec BSA processor.
\begin{figure}\centerline{ \epsfxsize=115mm \epsffile[22 162 836 533]{plaatjes/bsa.eps} } \end{figure}
The output signal of the photomultiplier is first amplified by a factor set by the operator and then bandpass filtered to remove frequency components outside the anticipated range of Doppler frequencies. A burst detection scheme determines whether the bandpass filtered signal contains a Doppler burst or not. Burst detection can be based on the pedestal or on the so called ``envelope.'' The envelope is obtained by rectifying and low-pass filtering of the bandpass filtered signal. When the envelope exceeds a 25 mV threshold, the sampler and the transit time counter are started and the arrival time is measured. The sampler is restarted each time the envelope exceeds the next higher threshold level (50 mV, 75 mV etc.). This is done to ensure that the samples are taken from the central part of the Doppler burst. The transit time counter is stopped when the envelope decreases below 12.5 mV.

While the burst detection scheme is carried out, the bandpass filtered signal is led through a mixer unit which shifts the power spectrum by a value of fc towards lower frequencies. The aim of this shift is to increase the resolution of the computed power spectrum. The centre frequency fc is selected by the operator in conjunction with the bandwidth Bw so that the cut-off frequencies of the bandpass filter are given by fc$ \pm$Bw/2. The down-shifted signal is low-pass filtered and then sampled at regular time intervals, tsam. The number of samples nrec is called the ``record length,'' and its value can be set by the operator at 8, 16, 32 or 64. The inverse of the time interval tsam is called the sampling frequency fsam. The resolution of the computed spectrum is proportional to fsam/nrec, which reduces to 1.5Bw/nrec because the BSA has a fixed relationship between the sampling frequency and the bandwidth: fsam = 1.5Bw. A hardwired FFT processor then computes a spectrum from the samples.

In the next step a sinc function is fitted to the computed spectrum at the frequency with the highest peak and its neighbouring frequencies. The Doppler frequency follows as the frequency for which the sinc function achieves a maximum. The thus determined Doppler frequency is validated by means of a comparison between the two highest peaks in the spectrum. The Doppler frequency is validated if the primary peak of the spectrum exceeds the secondary peak by a factor of 4 or higher. After validation the Doppler frequency together with the arrival time (optional) and the transit time (optional) are transferred to a computer.

Unlike, for example, TSI counter processors, the BSA processors cannot carry out a time-coincidence test to ensure that the measured Doppler frequencies originate from the same particle in case of a multi-component measurement. However, the time-coincidence test can be performed in the software that is used to reduce the raw data, provided that for each processor the arrival times of the particles are stored on disk. Alternatively, the different BSA processors can be run in the so called ``hardware-coincident mode.'' This mode of operation and its consequences are discussed later in this chapter.


The Three-Component LDA

Introduction

The interest of fluid-dynamics researchers for the three-component LDA (3-D LDA) is clear, because turbulence is a three-dimensional phenomenon and in many industrial flows even the mean flow is three-dimensional. In its early stages of development the 3-D LDA was notorious as far as the measurement accuracy of turbulence statistics was concerned. An increasing number of researchers came to the conclusion that the simultaneous measurement of three velocity components involved much more than bearing the financial burden for adding one LDA channel to an existing two-component system. The 3-D LDA poses a set of problems that are unique to this instrument. Meyers, a recognized expert in the field, sketched the development of the 3-D LDA in a paper entitled ``The Elusive Third Component'' [#!Meyers85!#]. Perhaps the title reflects the many problems encountered during the search for the right optical arrangement for the instrument. This section intends to discuss these problems and their remedies, thereby resulting in the following optical arrangement for the 3-D LDA: The discussion of the 3-D LDA will be limited to the dual-beam configuration, because of its superior signal-to-noise ratio.

Orthogonality requirement

Each channel of the 3-D LDA measures the velocity component in a direction that is determined by the orientation of the corresponding beam pair in space. In general, these primary or colour components are non-orthogonal and they do not coincide with one of the cartesian velocity components u, v or w. Therefore, the primary velocity components that are measured by the 3-D LDA must be transformed into the cartesian coordinate system. In this section the propagation of uncertainties in the primary velocities into the cartesian velocity components is investigated. The analysis of the transformation matrix will show that the orientation of the three beam pairs should be such that the primary velocity components are as close to orthogonal as possible.

Figure 1.7 : The orientation of a primary velocity component.
\begin{figure}\centerline{ \epsfxsize=68mm \epsffile[130 111 686 527]{plaatjes/third.prn} }\end{figure}
Figure 1.7 portrays a primary velocity component vp that is measured by one of the LDA channels. The orientation of this velocity component in the x, y, z-coordinate system is given by the angles $ \alpha$ and $ \beta$. A 3-D LDA gives rise to the following set of equations:
vg = u cos$\displaystyle \beta_{g}^{}$cos$\displaystyle \alpha_{g}^{}$ + v cos$\displaystyle \beta_{g}^{}$sin$\displaystyle \alpha_{g}^{}$ + w sin$\displaystyle \beta_{g}^{}$ (1.23)
vb = u cos$\displaystyle \beta_{b}^{}$cos$\displaystyle \alpha_{b}^{}$  + v cos$\displaystyle \beta_{b}^{}$sin$\displaystyle \alpha_{b}^{}$  + w sin$\displaystyle \beta_{b}^{}$  
vv = u cos$\displaystyle \beta_{v}^{}$cos$\displaystyle \alpha_{v}^{}$ + v cos$\displaystyle \beta_{v}^{}$sin$\displaystyle \alpha_{v}^{}$ + w sin$\displaystyle \beta_{v}^{}$ ,  

where u, v and w are the components of the velocity in the orthogonal coordinate system and vg, vb and vv are the primary velocity components measured by the green, blue and violet LDA channels, respectively. If, for reasons of simplicity, it is assumed that the blue and green channels form an orthogonal two-component LDA ( $ \alpha_{b}^{}$ = $ \alpha_{g}^{}$ - $ \pi$/2) that senses velocity components in the xy-plane ( $ \beta_{g}^{}$ = $ \beta_{b}^{}$ = 0) then Eq (1.23) reduces to
vg = u cos$\displaystyle \alpha_{g}^{}$ + v sin$\displaystyle \alpha_{g}^{}$ (1.24)
vb = u sin$\displaystyle \alpha_{g}^{}$ - v cos$\displaystyle \alpha_{g}^{}$  
vv = u cos$\displaystyle \beta_{v}^{}$cos$\displaystyle \alpha_{v}^{}$ + v cos$\displaystyle \beta_{v}^{}$sin$\displaystyle \alpha_{v}^{}$ + w sin$\displaystyle \beta_{v}^{}$ .  

Without loss of generality it may also be assumed that $ \alpha_{v}^{}$ = $ \pi$/2, so that the third component, w, can be expressed in terms of the primary velocities as

w = $\displaystyle {\frac{{\cos \alpha_g}}{{\tan \beta_v}}}$ vb - $\displaystyle {\frac{{\sin \alpha_g}}{{\tan \beta_v}}}$ vg + $\displaystyle {\frac{{v_v}}{{\sin \beta_v}}}$ . (1.25)
This equation shows that the coefficients of the primary velocities become large for small values of $ \beta_{v}^{}$. This causes the third component w to be very sensitive to uncertainties in the measured primary velocities caused by, for example, calibration errors or processor inaccuracies. In case the third component is measured directly, i.e. $ \beta_{v}^{}$ = $ \pi$/2, this extreme sensitivity is absent. So, ideally the transmitting optics of the 3-D LDA should be configured such that the device senses nearly-orthogonal velocity components.

In a more detailed analysis of the coordinate transform, Morrison et al. Morris90 showed that the uncertainty propagation into the third component is even more severe for higher-order statistics, such as the Reynolds stress $ \overline{{w'^2}}$, than it is for the mean velocity $ \overline{{w}}$. They conclude that the tilt angle $ \beta_{v}^{}$ should be at least 30o to keep the error propagation within reasonable limits. This requirement on the tilt angle poses a number of practical problems. Because many researchers do not know how to solve these problems (or are simply unaware of the orthogonality requirement), most operational 3-D LDAs are of the non-orthogonal type with small $ \beta_{v}^{}$. The practical problems are as follows. First, a large tilt angle requires optical access to the experimental facility from two adjacent sides which is difficult to realize in many existing wind tunnels. The second problem has to do with the alignment of the three beam pairs. The conventional procedure to align the beam pairs involves either a small pinhole or a microscope objective [#!Absil95!#]. Both methods can still be applied to the 3-D LDA as long as the tilt angle $ \beta_{v}^{}$ remains small, say $ \beta_{v}^{}$ < 15o, but they cannot be used for larger tilt angles. Consequently, the orthogonal 3-D LDA requires a new alignment procedure.

Virtual particles

In a study of the accuracy of a 3-D LDA, Boutier et al. Boutier85 found that some of the measured Reynolds stresses were systematically high, due to a phenomenon that they called ``virtual particles.'' The phenomenon is a consequence of the fact that any 3-D LDA has at least one measuring volume that does not fully overlap the other two. Only partial overlap of the measuring volumes can be achieved because the different optical axes cannot all coincide in 3-D LDA. This is in contrast to the two-component LDA where both measuring volumes usually share a single optical axis. The typical situation for a 3-D LDA is sketched in Fig. 1.8 where the optical axes of measuring volumes A and B include an angle $ \beta$. One of these measuring volumes actually consists of two fully overlapping volumes that is formed by two beam pairs (but that is not essential here.) Assume that each measuring volume senses a velocity component that lies in the plane spanned by the optical axes of measuring volumes A and B.
Figure 1.8 : The virtual-particle phenomenon in 3-D LDA.
\begin{figure}\centerline{ \epsfxsize=46mm \epsffile[201 102 586 522]{plaatjes/boutier1.prn} } \end{figure}

Now consider the following ``multiple-particle'' event. Volume A measures a particle with velocity component va at time ta whereas a particle with velocity component vb is measured by volume B at time tb. To verify whether the measurements on the two LDA channels stem from a single particle, it is common to apply a simultaneity criterion. In other words: if the arrival times ta and tb satisfy the criterion | ta - tb| < $ \tau_{w}^{}$, where $ \tau_{w}^{}$ is a user-selected time-coincidence window, then it is assumed that both measurements stem from a single particle. The LDA subsequently produces the velocity pair (va, vb) as if it represents the velocity components of a single particle. However, in the case of the multiple-particle event sketched in Fig. 1.8, the arrival times ta and tb may satisfy the simultaneity criterion, but they do not originate from the same particle. As a result, a ``virtual particle'' with velocity components (va, vb) is created, which will cause erroneous velocity statistics.

Boutier reasoned that the virtual-particle phenomenon was a complicated function of the tilt angle $ \beta$, the time-coincidence window $ \tau_{w}^{}$, the local flow conditions and the seed density. However, a solution to the problem was not given. Intuitively, it is clear that lowering the seed density will decrease the probability that virtual particles will occur, but it will not eliminate the problem. The only sensible way to circumvent the virtual-particle phenomenon is to collect data only from the region in space that is common to all (three) measuring volumes, which can be achieved by the positioning of small pinholes in front of the photomultipliers in conjunction with a large (near 90o) off-axis light-collection angle. This ``spatial filtering'' also happens to be the remedy for the geometry-bias problem that will be discussed below.


Geometry bias

In an attempt to quantify the findings of Boutier's investigation, Brown Brown89 simulated the operation of a typical 3-D LDA using a Monte-Carlo approach. The results of this study confirmed the existence of the virtual-particle phenomenon, and showed that, as expected, the probability of a virtual-particle occurrence increases with increasing seed density. Recall from the previous section that the virtual particles were able to pass the simultaneity criterion, thereby causing erroneous velocity statistics. Brown's study showed that even without virtual particles, which was easy to realize in the simulation, the velocity statistics as measured by the 3-D LDA were in error. As a result, the study revealed a previously unidentified error source. This error was termed the ``geometry bias,'' and it is a direct result of the 3-D LDA measuring-volume geometry in conjunction with the concept of a time-coincidence window.

Figure 1.9 : The geometry bias in 3-D LDA.
\begin{figure}\centerline{ \epsfxsize=45mm \epsffile[218 79 603 527]{plaatjes/brown1.prn} } \end{figure}
\begin{figure}\centerline{ \epsfxsize=31mm \epsffile[289 85 556 530]{plaatjes/brown2.prn} } \end{figure}
Figure 1.9 depicts the measuring-volume geometry that was used in Brown's study. The geometry is identical to that shown in Fig. 1.8 for a tilt angle $ \beta$ = 60o. Consider particle a that passes through the overlap region of the measuring volumes. For simplicity it is assumed that its velocity component in the y-direction is zero. Clearly, this particle will satisfy the time-coincidence criterion regardless of the magnitude of the velocity components u and w. This is not the case for particles b and c which do not pass through the overlap region. Particle b is assumed to have zero w-component and it will satisfy the time-coincidence criterion only if the in-plane velocity component u is sufficiently large. Particle c is supposed to have a non-zero w-component and it cannot pass the time-coincidence test if the w-component is large compared to the u-component, simply because it will not arrive at the other measuring volume. This illustrates that the 3-D LDA measuring-volume geometry in combination with the time-coincidence window will cause a bias towards high in-plane velocity components and small out-of-plane velocity components.

The time-coincidence concept, which works very satisfactorily for a conventional two-component LDA, is inadequate for the 3-D LDA. To circumvent the geometry bias, Brown Brown89 suggested a new mode of operation for the LDA signal processors known as the ``channel-blanking mode'' or the ``hardware-coincident mode.'' In this mode of operation each signal processor will process a Doppler burst only when Doppler bursts are also present on the other two channels, in the sense that the three Doppler bursts (partially) overlap in time. If this is not the case, the signal processors are inhibited. Due to the hardware-coincident mode, data will be acquired only from the overlap region of the three measuring volumes, so that particle a will be measured by the 3-D LDA while particles b and c are ignored1.1.

The hardware-coincident mode removes the geometry bias, which is a single-particle event. But it does not eliminate the virtual-particle phenomenon, because this is a multiple-particle event. To eliminate both error sources, the 3-D LDA requires both the channel-blanking mode and the collection of scattered light from the overlap region only, as mentioned in the previous section. The beneficial effect of these measures is that the spatial resolution of the 3-D LDA is high compared to that of a conventional two-component LDA. The latter is usually operated in the (off-axis) forward-scatter or backward-scatter mode, resulting in a sensitive region with relatively large dimensions. The sensitive region for the 3-D LDA is reduced to the overlap region of the three measuring volumes. This more-or-less spherical region has a characteristic length equal to the diameter of the individual measuring volumes which is typically 10 times smaller than the length of the measuring volumes. On the other hand, the smaller measuring volume of the 3-D LDA will result in a much lower mean data rate as compared to the two-component LDA for the same seed density.