VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS

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9TH. INTERNATIONAL SYMPOSIUM ON FLOW VISUALIZATION, 000 VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS Hui LI, Hui HU, Toshio KOBAYASHI, Tetsuo SAGA, Nobuyuki

1 9TH. INTERNATIONAL SYMPOSIUM ON FLOW VISUALIZATION, 000 VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS Hui LI, Hui HU, Toshio KOBAYASHI, Tetsuo SAGA, Nobuyuki TANIGUCHI Keywords: digital image processing, lobed mixing jet, multiresolution analysis, multiscale vortex, two-dimensional orthogonal wavelet transform ABSTRACT The two-dimensional orthogonal wavelet transform was applied to the LIF image of lobed mixing jet for identifying the multi-scale turbulent structures. The digital imaging slice photographs at x/d=6 and 4 with Re=3000 were respectively decomposed into seven image components with different broader scales. These image components provide visualized information on the multiscale structures in a lobed mixing turbulent jet. The coherent structure and edges of the vortices at different resolutions or scales can be easily extracted. It was found that the scale range of the coherent structure becomes small along the downstream direction, and the size of the intermediateand small-scale structures does not vary significantly with downstream distance. INTRODUCTION The lobed mixing jets are often used as an important device in the fluid machinery for their efficient mixing enhancement performance. It is now a well-known fact that the lobed mixing jet exhibited complex structure with a wide range of coexisting scales and a variety of shapes in the dynamics, and its coherent structures were responsible for most of the momentum, mass and heat transfer. Many identification techniques, such as image processing, spectra analysis, spatial correlation functions, proper orthogonal decomposition, stochastic estimation, and pattern recognition, were well established to determine the complex structure. However, these traditional analytical techniques cannot provide us sufficient or detail information. The complex turbulent structure in terms of space, scale and strength has not yet been clarified. To solve these problems, the wavelet analysis was applied to the complex turbulent structures since 989. The great progress of the wavelet applications in the fluid mechanics has been made. Li et al. [], [] employed one-dimensional continuous and discrete wavelet transforms to analyze the experimental velocity signals of turbulence in the dimensions of time and frequency. Li [3]-[6] proposed the wavelet correlation method and wavelet spatial statistics based on wavelets, and revealed the multi-scale structure of eddy motion in the turbulent shear flow in both Fourier and physical spaces. These researches indicated that the wavelet technique offered the potentials extracting new information from the turbulent field, however, they limited to the analysis of Author(s): Department of Mechanical Engineering Kagoshima University, Japan. nd Department, Institute of Industrial Science University of Tokyo, Japan. Corresponding author: Hui Li li@mech.kagoshima-u.ac.jp Paper number

2 Hui LI et al. turbulent structure based on the one-dimensional wavelet transform. To gain deeper insight into multi-scale structures and coherent structures in turbulent flows, it is an important to analyze the full field by the image processing. Although there were several researches [7]-[] that applied two or three-dimensional wavelet transform to full field measurements or simulation data, they have thus far concerned the continuous wavelet transform. In spite of that the coefficients of continuous wavelet transform may extract the characterization of local regularity, it is unable to reconstruct the original function because the mother wavelet function is a non-orthogonal function. In the turbulent image processing it is of great significance to study the image components of various scales that can reconstruct the original image based on the orthogonal wavelet transform. Recently, Li et al. [3] developed an application of twodimensional orthogonal wavelets to the turbulent images for the identification of the multiresolution turbulent structures. The aim of this paper was to apply the two-dimensional orthogonal wavelets, i.e. multiresolution analysis, to the digital imaging photographs of lobed mixing jets in order to reveal the multi-scale turbulent structures and to extract the most essential scales governing lobed mixing turbulent jet. TWO-DIMENSIONAL ORTHOGONAL WAVELETS For a two-dimensional scalar field f ( ) defined by x, x, the two-dimensional discrete wavelet transform is i j ( x, x ) Wf m n m n = f ( x, x ) Ψm n m n. (), ;, i j i j, ;, The reconstruction of the original scalar field can be achieved by using ( x, x ) = Wf Ψ ( x x ) f. () m, n ; m, n m, n ; m, n, m m n n The two-dimensional wavelet basis, Ψ ( x, x ) m, n; m, n functions generated by two one-dimensional bases as Ψ m, n; m, n, is simply to take the tensor product ( m ) ( ) + m m, ( m x x = x n ) ψ ( x n ) The oldest example of a function ψ () x for which the () x ψ. (3) ψ constitutes an orthogonal basis is the Haar function, constructed long before the term wavelet was coined. In the last ten years, various orthogonal wavelet bases have been constructed, for example, Meyer basis, Daubechies basis, Coifman basis, Battle-Lemarie basis, Baylkin basis, and spline basis, etc.. They provide excellent localization properties both in physical space and frequency space. In this study we use the Daubechies basis with index N=0, which is not only orthonormal, but also have smoothness and compact support, to analyze the flow image. The procedure of the wavelet analysis, i.e. multiresolution analysis, can be summarized in two steps: () Wavelet coefficients or wavelet spectrum of an image is computed based on the discrete wavelet transform of Eq. (). () Inverse wavelet transform of Eq. () is applied to wavelet coefficients at each wavelet level, and image components are obtained at each level or scale or in the wavelet spaces. The detail regarding the multiresolution analysis can be found in Li et al. [3]. m,n 9 th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS 3 EXPERIMENTAL PROCEDURE The experiment was carried out in liquid-phase turbulent-jet flows, the pulse laser sheet

The images of transverse sections at downstream position z/d=6 and 4 (jet-nozzle diameter d is.54mm) were captured by a 008x06 pixels Cross- Correlation CCD array camera.

3 VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS 3 EXPERIMENTAL PROCEDURE The experiment was carried out in liquid-phase turbulent-jet flows, the pulse laser sheet (thickness is about.0mm) used. Images of slices that relied on laser-induced fluorescence (LIF) digitalimaging techniques were obtained. The images of transverse sections at downstream position z/d=6 and 4 (jet-nozzle diameter d is.54mm) were captured by a 008x06 pixels Cross- Correlation CCD array camera. A lobed nozzle with six lobes was used, and the equivalent diameter is D=40mm. The height of the lobes is about 5mm, and the inner and outer penetration angles of the lobe structures are about 0 and 4 0, respectively. The jet velocity was set as about 0.m/s, and the Reynolds number based on the equivalent nozzle diameter was about 3,000. The field of view spans l0 cm, resulting in a pixel resolution of λ p 0µ m. More details have been given in Hu et al. [4]. 4 RESULTS AND DISCUSSION An original diametrical slice image of a lobed mixing jet with Re=3000 at downstream position z/d=6 is shown in Fig. (a). It is evident that the large-scale vortices in the shear layer dominate the whole flow structure. The regions of the higher strain (edges of the vortices) are identified as the higher gradient of concentration. To extract the multi-scale turbulent structures, the multiresolution analysis is employed to decompose this image. The wavelet coefficients of the original image are first computed based on the two-dimensional orthogonal wavelet transform. Then, the image components at seven different broader scales are reconstructed from the wavelet coefficients based on the inverse wavelet transform, and are displayed in Fig. (b)~(h). In Fig. (b)~(h), false-colors have been assigned to the scalar values of image components, and the highest concentration is displayed as a deep red and the lowest as purple. These images can provide information on the multi-scale structures in a lobed mixing turbulent jet and several features of the turbulent structure may be immediately noted in Fig.. At the smallest scale of a=0.4mm shown in Fig. (b), the whole field appears to be covered by a nearly homogeneous distribution of very fine scale turbulent structures. At the central scales of a=0.48mm and 0.96mm, as shown in Fig.(c) and (d), a clear distribution of the edges of the vortices with the smaller-scale can be observed in the interior of the flow except the center region. This means that the smaller-scale vortices exist in the whole flow field. The horizontal noisy lines appeared in the image components are caused by the background noise of the original image. As the central scale increases to (a) Original image (b) a=0.4mm Fig.. Multiresolution images of a lobed mixing jet at z/d=6 9 th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

4 Hui LI et al. (c) a=0.48mm (d) a=0.96mm (e) a=.9mm (f) a=3.84mm (g) a=7.68mm (h) a=5.36mm Fig.. Multiresolution images of a lobed mixing jet at z/d=6 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

5 VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS a=.9mm in Fig. (e), the edges of the vortices within this scale range can be clearly observed, which are the zoom-out image of a=0.96mm. At larger central scales of a=3.84mm and 7.68mm shown in Fig. (f) and (g), the core of the vortex is seen as a region of high negative concentration distributed in the shear layer. The higher concentration appearing in these two image components indicates that the vortices within these two broader scales are more active and dominate the turbulent mixing process, which are referred to as the coherent structure of the problem. The horizontal noisy lines existing in Fig. (a)~(e) are reduced within the larger scale range. At the largest scale of a=6.36mm (Fig. (h)), the large-scale flow field may be clearly identified by the green region. By comparing the original image, the positive and negative peaks indicate the position of large-scale vortices except two large positive peaks on the center. Figure (a) shows an original cross slice image of a lobed mixing jet with Re=3000 at downstream position z/d=4. The multi-scale vortices can be clearly seen in the original image. The multiresolution images at seven different broader scales are shown in Fig. (b)~(g). Within the range of small and intermediate scale of a=.9~0.48mm, the flow structures at z/d=4, as shown in Fig. (b)~(e), is similar to that at z/d=6. This indicates that the size of the intermediate- and small-scale structures does not vary significantly with downstream distance z. A clear distribution of the edges of the vortices in Fig. (e) can also be observed in the interior of the flow except the center region. However, at central scales of a=3.84mm the concentration of the vortex cores, as shown in Fig. (f), becomes lower than that in Fig. (f). The higher concentration of the vortex cores appearing in Fig. (g) indicates that the vortices within this broader scale are more active and dominate the turbulent mixing process, i.e. the coherent structure. This implies that the scale range of the coherent structure becomes small as increasing the downstream distance z. At the largest scale of a=6.36mm (Fig. (h)), the large-scale flow field, which is similar to that in Fig. (h), may also be clearly identified by the green region. 5 CONCLUSION Major conclusions are summarized as follows: () The multiresolution images within different broader scales can be obtained using the twodimensional orthogonal wavelets and provide information on the multi-scale turbulent structures in a lobed mixing jet. () The core and edge of the vortex can be identified at different broader scales. (3) The size of the intermediate- and small-scale structures does not vary significantly with downstream distance. (4) The scale range of the coherent structure becomes small along the downstream direction. REFERENCES [] Li H. and Nozaki T.: Wavelet Analysis for the Plane Turbulent Jet (Analysis of Large Eddy Structure). JSME International Journal, Fluids and Thermal Engineering, Vol.38, No.4, pp.55-53, 995. [] Li H., Takei M., Ochi M., Saito Y. and Horii K.: Eduction of Unsteady Structure in a Turbulent Jet by using of Continuous and Discrete Wavelet Transforms. Transactions of the Japan Society for Aeronautical and Space Sciences Vol.4, No.38, pp.0-7, 000. [3] Li H.: Application of Wavelet Cross-Correlation Analysis to a Turbulent Plane Jet. JSME International Journal, Fluids and Thermal Engineering, Vol.40, No., pp.58-66, 997. [4] Li H.: Wavelet Auto-Correlation Analysis Applied to Eddy Structure Identification of Free Turbulent Shear Flow. JSME International Journal, Fluids and Thermal Engineering, Vol.40, No.4, pp , 997. [5] Li H.: Identification of Coherent Structure in Turbulent Shear Flow with Wavelet Correlation Analysis. ASME Journal of Fluids Engineering, Vol.0, No.4, pp , 998. [6] Li, H.: Wavelet Statistical Analysis of the Near Field Flow Structure in a Turbulent Jet. Transactions of the Japan Society for Aeronautical and Space Sciences, Vol.4, No.33, pp.3-39, 998. [7] Everson R. and Sirovich L.: Wavelet Analysis of the Turbulent Jet. Phys. Lett., Vol.45, No.6, pp.34-3, th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

Hui LI et al. [8] Brasseur J. G. and Wang Q.: Structural Evolution of Intermittency and Anisotropy at Different Scales Analyzed Using Three-dimensional Wavelet Transforms. Phys. Fluids A, Vol.4, No.

0, pp.55-77, 993. [0] Dallard, T. and Browand, F.K.: Scale Transitions at Defect Sites in the Mixing Layer: Application of the -D Arc Wavelet Transform. J. Fluid Mech., Vol.47, pp.339-368, 993.

: The Education of Structure from Flow Imagery Using Wavelets (Part. The Mixing Layer). Experiments in Fluids, Vol.7, pp.67-74, 999. [3] Li H., Takei M., Ochi M., Saito Y. and Horii K.

0-7, 999. [4] Hu, H., Kobayashi T., Saga T., Segawa, S. and Taniguchi N.: PIV and LIF Measurements on the Lobed Jet Mixing Flows.

6 Hui LI et al. [8] Brasseur J. G. and Wang Q.: Structural Evolution of Intermittency and Anisotropy at Different Scales Analyzed Using Three-dimensional Wavelet Transforms. Phys. Fluids A, Vol.4, No., pp , 99. [9] Spedding G.R., Browand F.K., Huang N.E. and Long S.R.: A -D Complex Wavelet Analysis of an Unsteady WindGenerated Surface Wave Field. Dynamics of Atmospheres and Oceans, Vol.0, pp.55-77, 993. [0] Dallard, T. and Browand, F.K.: Scale Transitions at Defect Sites in the Mixing Layer: Application of the -D Arc Wavelet Transform. J. Fluid Mech., Vol.47, pp , 993. [] Dallard T. and Spedding G.R.: -D Wavelet Transform: Generalisation of the Hardy Space and Application to Experimental Studies. Eur. J. Mech. B/Fluids, Vol., pp.07-34, 993. [] Kailas S. V. and Narasimha R.: The Education of Structure from Flow Imagery Using Wavelets (Part. The Mixing Layer). Experiments in Fluids, Vol.7, pp.67-74, 999. [3] Li H., Takei M., Ochi M., Saito Y. and Horii K.: Application of Two-dimensional Orthogonal Wavelets to Multiresolution Image Analysis of a Turbulent Jet, Transactions of the Japan Society for Aeronautical and Space Sciences Vol.4, No.37, pp.0-7, 999. [4] Hu, H., Kobayashi T., Saga T., Segawa, S. and Taniguchi N.: PIV and LIF Measurements on the Lobed Jet Mixing Flows. Proceedings of the Third International Workshop on Particle Image Velocimetry, USA, pp.307-3, 999. (a) Original image (b) a=0.4mm (c) a=0.48mm (d) a=0.96mm Fig.. Multiresolution images of a lobed mixing jet at z/d=4 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

7 VISUALIZATION OF MULTI-SCALE TURBULENT STRUCTURE IN LOBED MIXING JET USING WAVELETS (e) a=.9 (g) a=7.68mm (f) a=3.84mm (h) a=5.36mm Fig.. Multiresolution images of a lobed mixing jet at z/d=4 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh,

APPLICATION OF WAVELET VECTOR MULTI-RESOLUTION TECHNIQUE TO PIV MEASUREMENTS

AIAA-- APPLICATION OF WAVELET VECTOR MULTI-RESOLUTION TECNIQUE TO PIV MEASUREMENTS ui LI Department of Mechanical Engineering Kagoshima University --, Korimoto, Kagoshima - JAPAN e-mail: li@mech.kagoshima-u.ac.jp