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• W569873619 abstract "A series of flow fields generated by a turbulent methane/air stratified swirl burner are investigated using laser Doppler anemometer (LDA). The LDA provides flow field measurements with comparatively high temporal resolutions. However, processing of the power spectral energy density (PSD) and autocorrelation functions (ACF) of the flow velocity by LDA is complicated by the random, intermittent nature of the LDA signal caused by random arrival of particles at the measuring volume. A tool is developed to overcome this difficulty and the preliminary results are presented in the present report. 1. Experimental Details The experimental details of the present report are provided in [4][7]. 2 Data processing 2.1 Data pre filtering The probability density function (PDF) of raw samples of axial velocity component u is shown in Figure 1a. Notice that there are small bumps at both ends of the PDF, which are considered to be noise. A 4 cut off filter is used to remove the noise. Figure 1. PDF distribution of velocity samples based on the raw data (a), and 4 filtered data (b). The profiles of mean axial velocities based on the raw and 4 filtered data are shown in Figure 2. The RMS of axial velocity converges to be more symmetric by applying the 4 filter. In -20 0 20 40 0 0.05 0.1 0.15 0.2 u (m/s) P ro b a b il it y (a) PDF(u) based on raw data -20 0 20 40 0 0.02 0.04 0.06 0.08 0.1 u (m/s) P ro b a b il it y (b) PDF(u) based on 4 filter 4 addition, the RMS peak at the location of each shear layer, i.e. r=±7, ±13 and ±18 mm, indicating that the current LDA setting can capture the general flow characteristics well. The 4 filter is adopted as a pre processing step in the tool for calculations of the moments and power spectral density. Figure 2. Filter test: profiles of mean and rms of the axial velocity based on the raw data (a), and 4 filtered data (b). 2.2 Transit time weighting During periods of higher velocity, a larger volume of fluid is swept through the measuring volume, and consequently a greater number of velocity samples will be recorded. As a direct result, an attempt to evaluate the statistics of the flow field using arithmetic averaging will bias the results in favor of the higher velocities. To correct this velocity-bias, a non-uniform weighting factor i  is introduced [2]. (1) 1 0     N j j i i t t  -where ti is the transit time or residence time of the i’th particle crossing the measurement volume. The transit time weighting is adopted in the tool for calculations of the moments. 2.3 Spectral analysis The definition of the autocorrelation function is   ( 2 ) ) ( ) ( E ) ( 2 1 t u t u Ru u      -20 -10 0 10 20 0 5 10 15 20 25 R (mm) (m /s ) (a) Raw data Raw u mean Raw u mean -20 -10 0 10 20 0 5 10 15 20 25 (b) 4 filtered data R (mm) (m /s ) Filtered u mean Filtered u mean -where   . . . E is the notation for the expectation value (mean). The sample times t1 and t2 are separated by the time delay 1 2 t t    . Thus, the autocorrelation function (ACF) ) ( uu R can be defined as: (3) ) ( ) ( 1 lim ) ( 0        T T uu dt t u t u T R   -where the integration limits T  0 reflects that real measurements always take place over a finite time span. The power density density (PSD) ) ( f Suu and the autocorrelation ) ( uu R form a Fourier transform pair: (4) ransform) (Fourier t ) ( ) ( 2          d e R f S f i uu uu (5) ansform) Fourier tr (Inverse ) ( ) ( 2     df e f S R f i uu uu    The correlation theorem states that the PSD can be calculated directly from the separate Fourier transforms:   (6) ) )U( U(E 1 lim ) ( f f T f S T uu    -where E [...] represent the expectation value and U(f) represent the finite Fourier transforms of ) (t u : ( 7 ) ) ( ) ( U 2 0 d t e t u f ft -i T     Since ) (t u is real, U(-f) equals U * (f), with U * representing the complex conjugate of U. The PSD can be estimated from U(f) directly: (8) ) U( ) ( U 1 ) ( ˆ f f T f Suu   Wiener-Khinchin Theorem states that: (9) ) U( 1 ) ( ˆ 2 f T f Suu  The definitions of correlations and spectra above are based on continuous signals ) (t u . In contrast to the straightforward computation of power spectral density functions for continuous or equally sampled data, LDA data is randomly sampled in time and no obvious equivalent to the fast Fourier transform is available as a computational algorithm. The present study used a sample-and-hold [1] reconstruction method to create equidistantly spaced time series, thereby allowing a FFT to be used in making PSD or ACF estimates. It can be written as (10) and for ) ( ) ( N ..., 3, 2, 1, = i , t < t t , t u = t u R +1 i i i  where NR is the total number of samples in a given block of data, u(ti) and u(t) are the true samples and resamples of the LDA data. The reconstruction can be performed either over the entire data set or within data blocks. The equidistant resampling with time steps of Δts is performed by (11) for ) ( N ..., 3, 2, 1, 0, = i , t i u = u R s i  and leads to a data set of NR samples that can be processed by a Fourier transform. The concept is well explained in Figure 3. Figure 3. Random true sample and S+H Resample. With resampled data, the integrals in equation (7) can be estimated as sums: (12) 2 exp ) ( U two of power a is if , of FFT 1 0          R R N u N n R s k N kn i u t f" @default.
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• W569873619 date "2012-06-08" @default.
• W569873619 modified "2023-09-23" @default.
• W569873619 title "A Tool for the Spectral Analysis of the Laser Doppler Anemometer Data of the Cambridge Stratified Swirl Burner" @default.
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