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9TH. INTERNATIONAL SYMPOSIUM ON FLOW VISUALIZATION, 2000 EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE Hui LI 1, Hui HU 2, Toshio KOBAYASHI 2, Tetsuo SAGA 2, Nobuyuki TANIGUCHI2 Keywords: orthogonal vector wavelet transform, particle image velocimetry, multiresolution analysis, multi-scale vortex, turbulent jet ABSTRACT The wavelet-based vector multiresolution technique was proposed and applied to the post-processing of PIV results for identifying the multi-scale turbulent structures. By analyzing the instantaneous streamlines and vorticity fields of various scales the important information on the multi-scale flow structures was obtained. The large-scale vortices or coherent structures with a central scale of a=15.36mm were easily extracted from PIV measurement results. The intermediate- and small-scale vortices embedded within the large-scale vortices were separated and visualized. 1 INTRODUCTION The turbulent jet exhibited a 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. During the last couple of years the development of PIV techniques has made it possible to provide more detailed information on the flow structures, such as the instantaneous values of various flow quantities, as well as their distribution and transient variation. Despite the usefulness of information were obtained by examining the measured instantaneous fields and the time-mean turbulent quantities, further information on the instantaneous multi-scale structures has not yet been clarified. In the past decade, there has been a growing interest in the use of wavelet analysis for the turbulent flow data [1]-[6]. This technique allows to track turbulent structures in terms of time and scale and extracts new information on turbulence structures. To gain deeper insight into the multiscale structures and coherent structures in the turbulent flows, it is an important to analyze the instantaneous full flow field. Recently, Li et al. [7], [8] firstly applied the two-dimensional orthogonal wavelets to the turbulent images for the identification of the multiresolution turbulent structures. However, few investigates concerned the extraction of multi-scale turbulent structures from PIV data. In this paper the wavelet vector multiresolution technique was developed and applied to the Author(s): 1 Department of Mechanical Engineering Kagoshima University, Japan. 2 2nd Department, Institute of Industrial Science University of Tokyo, Japan. Corresponding author: Hui Li E-mail: [email protected] Paper number 383 383-1 Hui LI et al. post-processing of PIV data for revealing and visualizing the instantaneous multi-scale turbulent structures. 2 WAVELET VECTOR MULTIRESOLUTION TECHNIQUE ! For a two-dimensional vector field f (x1 , x 2 ) and a wavelet basis Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ) the twodimensional discrete wavelet transform is defined by Wf m1 ,n1 ;m2 ,n2 = ∑∑ i ( ) " f ( x1i , x 2j )Ψ m1 ,n1 ;m2 ,n2 x1i , x 2j . (1) j The reconstruction of the original vector field can be achieved by using ! f (x1 , x 2 ) = ∑∑∑∑ m1 m2 n1 Wf m1 ,n1 ;m2 ,n2Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ) . (2) n2 The two-dimensional wavelet basis, Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ), is simply to take the tensor product functions generated by two one-dimensional bases as ( )( ) Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ) = 2 −(m1+m2 ) 2ψ 2 − m1 x1 − n1 ψ 2 − m2 x2 − n2 . (3) The oldest example of a function ψ (x ) for which the ψ m ,n (x ) 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=20, which is not only orthonormal, but also have smoothness and compact support, to analyze the flow image. The procedure of the vector multiresolution analysis can be summarized in two steps: (1) The wavelet coefficients Wf m1 ,n1 ;m2 ,n2 , representing vector field in orthonormal wavelets basis, are computed using the discrete wavelet transform of Eq. (1). (2) Inverse wavelet transform of Eq. (2) is applied to wavelet coefficients at each wavelet level, and components of vector field are obtained at each level or scale. 3 EXPERIMENTAL PROCEDURE The experiment was carried out in liquid-phase turbulent-jet flows. The PIV measurement was supplied by a double-pulsed Nd:YAG Laser at the frequency of 10 Hz and power of 200 mJ/pulse. The time interval between the two pulses can be adjustable, which is about 2 to 5 ms for the present study. The PIV images of transverse sections at downstream position z/d=12 (jet-nozzle diameter d is 2.54mm) were captured by a 1008x1016 pixels Cross-Correlation CCD array camera. The spatial resolution of the PIV images for the present research case is about 120 µm pixel . The images were divided into 8 by 8 pixel interrogation windows, and 50% overlap grids were employed. The Hierarchical Recursive PIV method was used in the present study for obtaining about 225x225 velocity vectors. The post-processing procedures which including sub-pixel interpolation and spurious velocity 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-2 EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE deletion were used to improve the accuracy of the PIV results. More details have been given in Hu et al. [9]. 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X Fig.1. Original instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (a) a=15.36mm Fig.2. Multiresolution instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-3 Hui LI et al. 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 800 1000 X (b) a=7.68mm 1000 800 Y 600 400 200 0 0 200 400 600 X (c) a=3.84mm Fig.2. Multiresolution instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000 4 RESULTS AND DISCUSSION Figure 1 showed an original instantaneous streamlines of a cross plane at downstream position z/d=12 in a lobed mixing jet with Re=3000. The irregular flow structures that imply a multi-scale 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-4 EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X Fig.3. Original instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000 structure can be observed. The irregular large-scale vortices, i.e. coherent structure, can be seen at the position of lobe. For identifying the multi-scale flow structures, the velocity vector filed obtained from the PIV measurement was decomposed into five components with different wavelet levels or broader scales based on the vector multiresolution analysis. The corresponding components of the streamlines were computed from the components of the velocity vector. Three components of instantaneous streamlines were shown in Fig.2. Figure 2 (a) exhibited clearly the large-scale structures with a central scale of a=15.36mm. The large-scale vortices, i.e. coherent structure, can be clearly seen at the position of lobe. These vortices corresponded quite well to the vortices in Fig.1. This agreement provides a validation for the present data analysis technique. The flow structures with a central scale of a=7.68mm can be shown in Fig.2 (b). A number of intermediate-scale vortices appeared at the position of lobe, and existed in the large-scale vortices of a=15.36mm (in Fig.2 (a)). It is clear that such structure cannot be extracted by traditional techniques. Figure 2 (c) showed the flow structures with a central scale of a=3.84mm. A number of smaller-scale vortices were identifiable. These vortices seem to occur all over the position of lobe. For even smaller-scale flow structures, it is difficulty to describe such flow structures using the instantaneous streamlines. We adopted the vorticity to discuss the vortical structures in the following session. Figure 3 illustrated a vorticity contour obtained from the original instantaneous PIV velocities, which corresponded to the instantaneous streamlines of Fig.1. The positive and negative vorticities were denoted by solid and dashed lines, respectively. Only a number of smaller peaks in the vorticity field that represented the smaller-scale vortices can be observed in Fig.3. In order to extract the distribution of the multi-scale vorticity field, the components of the velocity vector obtained using the wavelet vector multiresolution technique were employed to compute the vorticity field. The components of vorticity contour with five broader scales were displayed in Fig.4, and provided information on the distribution of multi-scale vortices in a lobed mixing turbulent jet. Several alternative larger positive and negative peaks can be clearly seen at the position of lobe in Fig.4 (a), and indicated pairs of large-scale streamwise vortices with a central scale of a=15.36mm. They are the uppermost and coherent structures. These larger positive and negative peaks consistent with the large-scale vortices observed in the instantaneous streamlines of Fig.2 (a). 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-5 Hui LI et al. 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (a) a=15.36mm 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (b) a=7.68mm Fig. 4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000 In the vorticity component of scale a=7.68mm (Fig.4 (b)), several pairs of positive and negative peaks appeared at the position of each lobe. They implied pairs of intermediate-scale streamwise vortices. By comparing the Fig.4 (a), it was found that each pair of intermediate-scale streamwise vortices was contained in a larger-scale vortex. With decreasing the scale to a=3.84mm, as shown in Fig.4(c), a number of stronger positive or negative peaks mainly distributed in 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-6 EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (c) a=3.84mm 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (d) a=1.92mm Fig.4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000 neighbor of lobe, and correspond to the vortices at this scale range. When further decreasing the scale to a=1.92 and 0.96mm, as shown in Fig.4 (d) and (e), a clear distribution of a number of smaller positive or negative peaks can be observed in the interior of the flow. This means that the smaller-scale vortices exist in the whole flow field. Note that it is difficulty to identify such multiscale flow structures by analyzing the original vector field of PIV results. 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-7 Hui LI et al. 1000 800 Y 600 400 200 0 0 200 400 600 800 1000 X (e) a=0.96mm Fig.4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000 Although preliminary, the results presented here demonstrated that the wavelet vector multiresolution technique could be used effectively for decomposition and analysis of multi-scale turbulent structures. 5 CONCLUSION In this paper the wavelet-based vector multiresolution technique was developed and applied to PIV data. The components of instantaneous streamlines and vorticity were obtained, and important information on the multi-scale flow structures was provided. The large-scale vortices or coherent structures with a central scale of a=15.36mm were extracted from PIV measurement results. The intermediate- and small-scale vortices embedded within the large-scale vortices were visualized. REFERENCES [1] Li H. and Nozaki T.: Wavelet Analysis for the Plane Turbulent Jet (Analysis of Large Eddy Structure). JSME [2] [3] [4] [5] [6] International Journal, Fluids and Thermal Engineering, Vol.38, No.4, pp.525-531, 1995. 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.42, No.138, pp.120-127, 2000. Li H.: Application of Wavelet Cross-Correlation Analysis to a Turbulent Plane Jet. JSME International Journal, Fluids and Thermal Engineering, Vol.40, No.1, pp.58-66, 1997. 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Transactions of the Japan Society for Aeronautical and Space Sciences, Vol.41, No.133, pp.132-139, 1998. 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-8 EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE [7] 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.42, No.137, pp.120-127, 1999. [8] Li H., Hu, H., Saga T., Kobayashi T., Taniguchi N. and Segawa, S.:Visualization of Multi-Scale Turbulnet Structure in Lobed Mixing Jet Using Wavelets. The proceedings of 9th International Symposium on Flow Visualization, U.K., No.195, 2000. [9] Hu, H., Saga T., Kobayashi T., Taniguchi N. and Segawa, S.: Improve the Spatial Resolution of PIV Result by Using Hierarchical Recursive Operation. to be published in Journal of Visualization. 9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000 Editors G M Carlomagno and I Grant. 383-9