Application of Wavelet Multiresolution Analysis to Jet Flow Issuing From Rotating Circular Pipe With Inclined Section

Abstract

The flow feature of the jet issuing from the circular pipe with the rotating inclined section has been investigated by the methods of flow visualization and image processing. It has been found that the jet diffusion is affected by the inclined angle and the rotating speed. The coherent structure of the jet has also been studied by using wavelet multiresolution analysis. The multiscale turbulent structures were visualized and the core and edge of the vortex were identified at different broad scales.

The X-Ray Transform Projection of 3D Mother Wavelet Function

“…Suppose ψ ( x ) ∈ L 2 ( R n ) is a complex-value function ( n is a positive integer); then ψ ( x ) is referred to as mother wavelet function (or wavelet base function) [21, 22] if the following admissibility condition is satisfied: $\begin{array}{c}{\int }_{R^n}\frac{{\left|\widehat{\psi }\left(\mathbf{w}\right)\right|}^{\mathrm{2}}}{{\left|\mathbf{w}\right|}^n}d\mathbf{w}=C_{\psi }<+\infty ,\end{array}$ where $\hat{\psi }false(\mathrm{bold-italicw}false)$ is the Fourier transform of ψ ( w ). Assuming ψ ( x ) ∈ L 1 ( R n ), then $\hat{\psi }false(\mathrm{bold-italicw}false)$ is continuous, and $\hat{\psi }false(\mathrm{bold-italicw}false)=\mathrm{normal0}$ according to admissibility condition (1).…”

“…In previous papers, we discussed the problem of inverse transformation; namely, the X-ray inverse transformation of 2D mother wavelet is 3D mother wavelet function under certain conditions. So, the 3D mother wavelet is structured by the back projection of 2D mother wavelet functions [21, 22]. In this paper, discussing the problem of normal transformation (the X-ray transformation of 3D mother wavelet is 2D mother wavelet function that satisfy certain conditions) and realizing 3D wavelet transformation through 2D wavelet of the multiangle projection.…”