Acoustic Particle Velocity Measurements Using Laser by Jean-Christophe Valière

Author:Jean-Christophe Valière
Language: eng
Format: epub
Publisher: Wiley
Published: 2014-03-25T04:00:00+00:00

Now, let us assume that da = Va/2πfa is much larger than β, i.e. Va above the line in Figure 3.9. We also consider in the present case that Va > vc and the turbulent fluctuations are small. Equation [3.21] becomes

[3.32]

As the acoustic displacement is much larger than the PV, the analysis time Δt = t − t0 is very short compared with the acoustic period (see also equation [2.16]). Then, t + t0 = 2t0 and πfaΔt << 1, equation [3.32] becomes

[3.33]

and then

[3.34]

This probability leads to a non-stationary Poisson process (NSPP) described in Chapter 2. It is possible to assume from the last equation [3.33] that the probability to detect particles is equal inside a time interval [t1,t2] modulo the acoustic period. Then, the probability of detecting a q particle on this interval is

[3.35]

where M(t) is the local density. On the interval [t1,t2], is the mean associated with a phase of the signal. Inside each interval, it is possible to consider the probability of particle detection as a Poisson process (equation [2.73]).

Concerning the signal itself, as the windowing is very short, the magnitude and frequency modulations could be considered as constant during its length. Then, applying the approximation on a short window on equations [3.12]–[3.14], the Doppler signal in this case may be reduced to

[3.36]

[3.37]

[3.38]

If Δt is short, the magnitude could be considered as constant during the window length. vf(t0) is the value of the fluctuating velocity during the window and Φ(t0) is the phase due to the initial condition (apparition of the particle). This last equation will give

[3.39]

and its analytical signal

[3.40]

with A(t) including the effect of noise. The instantaneous frequencies (equation [3.29]) give

[3.41]

and do not depend on time. So each particle naturally windowed by the Gaussian shape gives only one measuring point. This point contains not only the acoustic velocity at time t0 but also the local hydrodynamic velocity (mean and turbulent), which may be estimated previously in order to improve the acoustic measurement.

A simulation has been achieved using the model presented in section 3.2.2. Figure 3.11 shows the result with few mean flows (vc = 0.01 m/s) and high acoustic level (Va = 5 m/s), fa = 100 Hz and a realistic level of noise. In Figure 3.11(a), 50 ms of the Doppler signal is displayed. Bursts are clearly generated randomly. In Figures 3.11(b) and (c), two bursts have been zoomed in upon. They have two different local frequencies that seem to be constant during the burst duration, proving equation [3.40].

Figure 3.11 Simulation of the Doppler signal with Va = 5 m/s, vc = 0.02 m/s and fa = 100 Hz. a) 50 ms of the Doppler signal; b) and c) zoom of two different bursts

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