Structure Evaluation of Unsteady Turbulent Flow with Continuous and Discrete Wavelet Transforms

To evaluate eddy structures of a plane turbulent jet in the dimension of time and scale, the velocity signals were analyzed using the continuous and discrete wavelet transforms in this paper. From the distribution of coefficients of continuous wavelet transform, localized nearly periodic eddy motions with a=64ms, 100ms and 180ms were respectively observed in the shear layer of x/d=8.5 in the time range of t=0~220ms, 280~600ms and 680~1000ms, although the local scale of motion always changes with time in unsteady turbulent jet. From multiresolution analysis or discrete wavelet transform, the peak that appears in the component of fluctuating velocity represents the passing of eddy through the shear layer and concentration of the energy of the flow at instant. The intermittent eddy phenomenon or zero components of fluctuating velocity can be observed at higher levels or smaller scales. INTRODUCTION The large-scale eddy motion in a plane turbulent jet exhibits an symmetric, periodic and apparent flapping motion in similarity region, and the evolution and interaction of largescale organized structures play an important role in a turbulent jet spreading and momentum transfer. Until now the conventional statistical methods, such as, space-time correlation functions, spectra, coherent functions, conditional sampling methods, and visualization techniques are wellestablished usual techniques for gaining information regarding the nature of turbulent structure or eddy motion. However, turbulence or eddy motion is characterized by unsteady and localized structure of multiple spatial scales. Some important spatial information is lost owing to the non-local nature of the Fourier analysis. The visualization of organized motions in shear layers also showed that the conditional sampling measurement had been hiding very important features of turbulence. In recent decade, there has been growing interest in the wavelet analysis of turbulent signal, which can combine timespace and frequency-space analyses to produce a potentially more revealing picture of time-frequency localization of turbulent structure. The wavelet transform can either be continuous or discrete, and yields elegant decompositions of turbulent flows. The continuous wavelet transform offers a continuous and redundant unfolding in terms of time and scale and thus can track coherent structures (Li and Nozaki, 1995; Li, 1998a) and discriminate different flow regimes (Li and Tomita, 1997). Besides these application studies, several new tools and diagnostics based on the wavelet transform, such as wavelet correlation function (Li, 1998a), wavelet Reynolds stress function (Li, 1998b), wavelet triple velocity correlation (Li, 1998b), and local wavelet Reynolds stress function (Li et al., 1997) were developed. They offered the potentials extracting new information from various flow fields. The coefficients of continuous wavelet transform can extract the 1 Copyright © 1999 by ASME characterization of local regularity, but it is unable to reconstruct the original function because the mother wavelet function is non-orthogonal function. In the signal processing, it is importance to reconstruct the original signal from wavelet composition and to study multiresolution signal in the range of various scales. The discrete wavelet transform allows an orthogonal projection on a minimal number of independent modes and is invertible and in fact orthogonal inverse transform. Such analysis is known as a multiresolution representation and might be used to compute or model turbulent flow dynamics. Li et al. (1998) applied the two-dimensional orthogonal wavelets to turbulent images, and extracted the multiresolution turbulent structures and the coherent structure. However, few investigations concerned the application of the discrete wavelet transform to turbulent signals. To evaluate the vortical structures in the dimension of time and scale in a turbulent jet, the continuous and discrete wavelet transforms were applied to the velocity signals of a plane turbulent jet in this paper. CONTINUOUS WAVELET TRANSFORM For any signal ( ) ( ) R ∈ p L t f ∞ ≤ ≤ p 1 ( ( ) R p L denotes the Hilbert space of measurable) the continuous wavelet transform can be defined as