Data analysis / reduction

The complexity of most aeroacoustic systems—being associated with high Reynolds number turbulence—means that we frequently find ourselves faced with the task of making sense of large quantities of data; such databases may be the result of numerical simulation and/or experimental measure­ments. Some form of data synthesis, or reduction, is necessary. The data can be considerably compressed, for example, by considering only the time – averaged values of the dependent variables, but at the loss of a large quantity of information. Other time-averaged statistical moments, such as the root mean square (2nd order moment), skewness (3rd order moment) and kur – tosis (4th order moment) can be computed—further information is thereby obtained regarding the state of the system.

Between such time-averaged quantities and the full space-time struc­ture of the system considered there lie many intermediate possibilities for compressing the data into manageable and insight-providing forms. Four techniques by which such intermediate data compression can be obtained (Fourier transform, Wavelet transform, Proper Orthogonal Decomposition and Dynamic Mode Decomposition) are presented in this section, example implementations being found in section §4. Further to these data com – pression/decomposition tools we also present a technique, known as Linear Stochastic Estimation, for the computation of conditional averages. This can constitute a powerful complementary approach when used in conjunc­tion with the said data compression/decomposition tools.

The four data compression techniques discussed have the following com­mon property: they all involve the expansion of space-time data in terms of sets of basis functions. The interest in such an operation is that the very high dimensional flow data can be broken down into a more manageable number of ‘building blocks’, conducive to perspicacious analysis and mod­elling. In the case of spectral and wavelet analyses, the basis functions are analytic and specified a priori; in the case of Proper Orthogonal Decompo­sition the basis functions are empirical and thus intrinsic to the data; in the case of Koopman modes (obtained by Dynamic Mode Decomposition), the functions are associated with the dynamics of the system, in other words they contain information regarding the temporal evolution of the system.

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