Estimation of burst-mode LDA power spectra

Abstract

The estimation of power spectra from LDA data provides signal processing challenges for fluid dynamicists for several reasons: acquisition is dictated by randomly arriving particles, the registered particle velocities tend to be biased toward higher values, and the signal is highly intermittent. The signal can be interpreted correctly by applying residence time weighting to all statistics and using the residence time-weighted discrete Fourier transform to compute the Fourier transform. A new spectral algorithm using the latter is applied to two experiments: a cylinder wake and an axisymmetric turbulent jet. These are compared with corresponding hot-wire spectra as well as to alternative algorithms for LDA signals such as the time-slot correlation method, sample-and-hold and common weighting schemes.

Appendices

Appendix 1: High-velocity and directional bias correction

It is clear that the method of residence time-weighted data processing recapitulated above provides the correct statistical results, since it was shown to be equivalent to conventional time averages. Thus, the concept of bias does not enter at all. However, the method is only correct for uniformly seeded flows of constant, uniform density. It was also assumed that no other sources of bias existed, e.g., directional bias due to the finite fringe number effect or electronic biasing.

The concept of bias in burst-type LDA measurement and methods of bias correction were introduced in 1973, when it became clear that the results of direct arithmetic averaging contained errors, so-called bias errors, especially for measurements in high intensity turbulence. One of the first treatments on the problem was given by McLaughlin and Tiederman (1973), who defined the problem and proposed a method of correction based on the weighting of the measured data, u ₀(t _i ), with the inverse of the numerical value of the measured velocity component, i.e., with the factors u ₀(t _i )⁻¹. The algorithms for the computation of mean and mean-square values by this method, from here on termed the one-dimensional or 1-D correction, are:

$$\overline{u({\bf x_0})}_{1\text{-}{\rm D}}=\frac{\sum\nolimits_i u_0(t_i) |u_0(t_i)|^{-1}}{\sum\nolimits_i |u_0(t_i)|^{-1}}$$

(24)

and

$$\overline{[u^{\prime}({\bf x_0})]^2}_{1\text{-}{\rm D}}=\frac{\sum\nolimits_i [u_0(t_i) - \overline{u}({\bf x_0})]^2 |u_0(t_i)|^{-1}}{\sum\nolimits_i |u_0(t_i)|^{-1}}$$

(25)

McLaughlin and Tiederman investigated the accuracy of the 1-D correction by numerical simulation of certain 2-dimensional flow cases. However, they did not consider three dimensional cases and did not consider the directional bias due to the finite fringe number effect. ⁶ Buchhave (1976) investigated the accuracy of l-D correction data by simulating data from isotropic turbulence, and included the directional bias. It was shown that the l-D correction over-compensates for the velocity bias and leads to large errors for turbulence intensities above 15–20 %. However, the angular bias, which occurs for many types of LDA counters used up to this time, pulls in the other direction, and under optimal conditions quite good results may be obtained in certain situations even for turbulence intensities in the order of 50 %. Figure 10 from (Buchhave 1976) shows the errors in computed mean and mean-square values as a function turbulence intensity of an isotropic, Gaussian turbulence superimposed on a constant mean velocity.

Fig. 10

Errors in computed mean and mean-square velocity as a function of turbulence intensity (taken from Buchhave 1976)

The problem of bias was discussed for the first time at the 1974 Purdue meeting (see, e.g., Durst 1974). The theory was further developed by George (1976) along the lines presented earlier in the current work. Hosel and Rodi (1977) also developed the theory from considerations on the average data sampling rate for varying velocities and provided some experimental results to support the theory. Later Erdman and Gellert (1976) studied the correlation between velocity and particle arrival rate and showed results which substantiate the ideas about proportionality between data rate and velocity in one-dimensional flows.

Some investigators, however, report a weaker correlation between velocity and data rate than expected from the preceding theory (see, e.g., Smith and Meadows 1974). In these cases electronic noise may have caused secondary effects, since these measurements were made at rather high velocities.

More recent measurements by Karpuk and Tiederman (1976) and Quigley and Tiederman (1977) in the viscous sublayer in pipe flow show good agreement between mean and rms values computed by the l-D correction method and hot-wire data. The measurements were made with an optical system especially designed to give a probe volume with small spatial dimensions in the direction of the mean velocity gradient. The authors note that the data rate did not seem to be correlated with direction, but unfortunately no details are given on the ratio of the number of fringes needed for operation of the signal processor to the total number available in a burst.

Finally, concerning velocity bias it may be mentioned (as already pointed out by McLaughlin and Tiederman) that in simultaneous measurement of all three velocity components (or in the case of two-dimensional measurements in flows in which the fluctuations in the third direction are negligible) it is of course possible to compute the magnitude of the velocity ${\bf u_i}$ and assign a weighting factor proportional to $|{\bf u_i}|^{-1}$ to each sample set (u _i , v _i , w _i ), and even to correct for directional bias based on the knowledge of the direction of u. Such corrections might be carried out on stored data points after the measurement, but it appears that this method has not yet been tried in actual measurements.⁷

The question of whether other factors influence the data rate is still largely unresolved. Tiederman (1977) raised the question of whether differences in signal-to-noise ratio of fast and slow bursts might influence the data rate. Other physical effects that cause correlation between data rate and flow velocity include density variations caused by pressure or temperature fluctuations, mixing of fluids with different particle concentration, and chemical reactions. Asalor and Whitelaw (1976) derived expressions for the correlation between combustion induced temperature, pressure and concentration fluctuations, and computed the data rate based on assumptions about the velocity-temperature and velocity-pressure correlations in a diffusion flame. From this analysis and subsequent measurements the authors concluded that in this particular flow the bias effects due to velocity fluctuations confirmed the velocity-data rate correlations expected from the residence time analysis and, e.g., Maclaughlin and Tiederman’s assumptions. Velocity-pressure and velocity-temperature correlation effects were found to be negligible.

George (1976) discusses briefly the extension of the arguments leading to the residence time weighting to flow with density fluctuations. No previous attempts seem to have been made on weighting or bias correction in LDA measurements of correlation functions or spectra.

The slotted-time-lag autocovariance method developed by Gaster and Roberts (1975) was simultaneously being developed by Mayo and others (see, e.g., Smith and Meadows 1974; Mayo 1974; Scott 1974). Curiously no one seemed to notice that the time slots themselves re-introduced the aliasing that was to have been eliminated by the random sampling. Measurements on the turbulence of a free jet were reported by Smith and Meadows (1974). The basic feasibility of measurements of turbulence power spectra with burst-type LDAs was proven. Later also other measurements were reported (Mayo et al. 1974; Bouis et al. 1977). However, none of these report any attempt to consider weighting of the data along the lines discussed above or to correct for biasing. Wang (1976) and Asher et al. (1974) discuss various sources of error and noise in LDA-counter measurements of power spectra and conclude that the quantizing of the output from the LDA-counter due to the finite resolution of the counter itself is the greatest source of error. However, these reports did not consider the “apparent turbulence” caused by the finite dimensions of the measuring volume in the presence of gradients within the volume as described in (Buchhave 1979), Chapter 4.5, nor did it consider the biasing effects introduced by using uncorrected data. It should be apparent that bias effects will modify the computed spectra as well as mean and mean-square values and that bias-correction methods should be applied to spectral measurements as well. The theoretical considerations presented earlier indicate that the phenomenon of bias need not exist in the sense that it had been considered here and results only from incorrect signal processing.

Appendix 2: Proposed Matlab-algorithm

Appendix 3: Errors from omitting residence time weighting

As was described by Velte (2009), the effect of computing the arithmetic statistical moments instead of the residence time weighted ones can be evaluated by performing a Taylor expansion. For both the mean value and the variance (which the power spectrum is directly linked to), a Taylor expansion has been performed about the state of zero turbulence intensity. It is assumed that all particles pass through the center of the measurement volume and traverse undisturbed. Further, only a one-dimensional flow is considered, say $\tilde{u} = U+u$, where U is the mean and u _n the departure of the nth realization from it.

It is important to note that, from the expansions, it is apparent that large values of higher-order moments can affect the accuracy significantly. This is usually not the case at the jet centerline, where the mean velocity is the largest and the velocity variance is the main higher-order term. But as one moves further and further off the centerline, the mean velocity decreases and the impact of the higher moments (skewness, kurtosis, etc.) increase, resulting in larger errors.

First, let’s consider the residence time weighted mean velocity:

$$U_{0,T}^{B} = \frac{\sum\nolimits_{n=0}^{N-1}\tilde{u}_n \Updelta t_n}{\sum\nolimits_{n=0}^{N-1}\Updelta t_n}$$

(26)

The above assumptions imply the following approximate form for the sum of residence times:

$$\sum\limits_{n=0}^{N-1}\Updelta t_n \approx \sum\limits_{n=0}^{N-1} \frac{d}{U+u_n} = \frac{d}{U}\sum\limits_{n=0}^{N-1} \frac{1}{1+u_n/U}$$

(27)

where d is the measuring volume length. Expanding the denominator yields

$$\sum\limits_{n=0}^{N-1}\Updelta t_n \approx \frac{d}{U}\sum\limits_{n=0}^{N-1} \;\; \left \{1 - \frac{u_n}{U} + \frac{u_n^2}{U^2} - \frac{u_n^3}{U^3} + \frac{u_n^4}{U^4} - \cdots \right \}$$

(28)

$$\approx N\frac{d}{U} - 0 + \frac{d}{U}\sum\limits_{n=0}^{N-1}\frac{u_n^2}{U^2} - \frac{d}{U}\sum\limits_{n=0}^{N-1}\frac{u_n^3}{U^3} + \frac{d}{U}\sum\limits_{n=0}^{N-1}\frac{u_n^4}{U^4} \approx N\frac{d}{U} \left \{ 1 + \frac{\overline{u^2}}{U^2} - \frac{\overline{u^3}}{U^3} + \frac{\overline{u^4}}{U^4}\right \}$$

(29)

Similarly, the Taylor expansion of the residence time weighted mean velocity can be expanded to second order in $\overline{u^2}/U^2$ to obtain:

$$U_{0,T}^{B} \approx \frac{Nd}{N\frac{d}{U}\left \{ 1 + \frac{\overline{u^2}}{U^2} \right \}} \approx U \left \{ 1 - \frac{\overline{u^2}}{U^2} + \frac{\overline{u^3}}{U^3} - \frac{\overline{u^4}}{U^4} + \cdots \right \}$$

(30)

Thus, the deviation of the arithmetic mean to the residence time weighted one is proportional to the variance and inversely proportional to the mean velocity.

$$\frac{\Updelta U_{0,T}^{B}}{U} = \frac{U_{0,T}^{B} - U}{U} \approx -\frac{\overline{u^2}}{U^2}$$

(31)

It is clear from the above results, which were confirmed for the data sets presented in section 4 in Velte (2009), that the turbulence intensity is crucial for low bias errors in the mean value.

The same procedure can be applied to the higher moments. Consider the second moment (which is vital in estimating the power spectrum), where

$$\overline{u^2}_{0,T}^{B} \approx \frac{d\sum\nolimits_{n=0}^{N-1}(U+u_n)}{N\frac{d}{U}\left \{ 1 + \frac{\overline{u^2}}{U^2} \right \}} \approx \frac{NdU}{N\frac{d}{U}\left \{ 1 + \frac{\overline{u^2}}{U^2} \right \}} \approx U^2 \left \{ 1 - \frac{\overline{u^2}}{U^2} + \left( \frac{\overline{u^2}}{U^2} \right)^2 + \cdots \right \}$$

(32)

which yields

$$\frac{\Updelta \overline{u^2}}{U^2} = \frac{\overline{u^2}^B_{0,T} - \overline{u^2}}{U^2} \approx 1-2\frac{\overline{u^2}}{U^2}+\left( \frac{\overline{u^2}}{U^2} \right)^2$$

(33)

and so on.