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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.
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 . In complex notation the plane wave is given byk0 = 2/ is the wave number. The wavelength is related to the frequency of the light source as = 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 , and the detector is located at = . The spherical wave emitted by the particle may be represented by [#!Adrian83!#]: ( = ) it follows from Eq (1.1) and Eq (1.2) that ( = )
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 10-13. In the low-velocity range, say < 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.
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 infD 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 - . 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!#]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 - 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 1 and Tsfw2 1) Ts, say Ts 10-9 s, Eq (1.10) reduces to the following well-known expression for the output signal of the photomultiplier fD.
1.2, it is easy to see that the expression for the Doppler frequency, Eq (1.7), can be rewritten as x-axis. The frequency difference fD can be positive or negative depending on the value of . However, the output of the photomultiplier cannot distinguish between positive and negative values of fD because cos(- fD) = cos(fD).
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
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 becomesvs = dffs in the positive y-direction. As a result, a detector sees intensity variations with a frequency 1.13), the result obtained using the Doppler theory.
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 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 m and fs = 40 MHz.
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)dp is the particle diameter, up and uf are the particle and fluid velocities and and 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 / 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 = eit and up = ()eit in Eq (1.21) yields the amplitude ratio || as1.22) 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 || = 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 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.
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 > 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.
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. 1.6.25 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 fcBw/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.
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.
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 and . A 3-D LDA gives rise to the following set of equations:
|u coscos + v cossin + w sin
|u coscos + v cossin + w sin
|u coscos + v cossin + w sin ,
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 , than it is for the mean velocity . They conclude that the tilt angle 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 . 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 remains small, say < 15o, but they cannot be used for larger tilt angles. Consequently, the orthogonal 3-D LDA requires a new alignment procedure.
1.8 where the optical axes of measuring volumes A and B include an angle . 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.
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| < , where 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 , the time-coincidence window , 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.
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.