APPLICATION OF WAVELET VECTOR MULTI-RESOLUTION TECHNIQUE TO PIV MEASUREMENTS

Transcription

1 AIAA-- APPLICATION OF WAVELET VECTOR MULTI-RESOLUTION TECNIQUE TO PIV MEASUREMENTS ui LI Department of Mechanical Engineering Kagoshima University --, Korimoto, Kagoshima - JAPAN li@mech.kagoshima-u.ac.jp ui U Turbulent Mixing and Unsteady Aerodynamics Lab. A, Research Complex Engineering Michigan State University East Lansing, Michigan USA Toshio KOBAYASI, Tetsuo SAGA, Nobuyuki TANIGUCI Institute of Industrial Science University of Tokyo -- Roppongi, Minato-Ku, Tokyo - JAPAN ABSTRACT A wavelet-based vector multi-resolution technique was applied to analyze the three-dimensional measurement results of a high-resolution stereoscopic PIV system in this paper. The instantaneous threedimensional flow structures in the near field of lobed jet mixing flow were successfully decomposed into large- and small-scale structures based on the wavelet vector multi-resolution analysis. It is found that as increasing the downstream distance, the large- and small-scale streamwise vortices and the higher smallscale w velocity component first grow up and appear around the trailing edge of the lobed nozzle, and then they decay rapidly and appear in the center region of jet. INTRODUCTION The turbulent jet exhibited complex structures with a wide range of coexisting scales and a variety of shapes in the dynamics and its physics of mixing process is important in the engineering. It is well known fact that the streamwise vortices generated in a jet flow, in additional to the azimuthal (or ring type) vortices, have been found to mix fluid streams even Copyright by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. more efficiently. The streamwise vortices in jet mixing flows can be generated by many methods. One of methods is to use a lobed nozzle to generate the largescale streamwise vortices, which has been considered to be one promising method for the jet mixing enhancement. During the last couple of years the development of PIV techniques has made it possible to provide more detailed information on flow structure, such as the instantaneous values of various flow quantities, as well as their distribution and transient variation. (), () Recently, u et al. () employed twodimensional and stereoscopic PIV system to measure the near flow field of a lobed jet mixing flow. The characteristics of the mixing process in a lobed jet mixing flow compared with a conventional circular jet flow were discussed based on the two and threedimensional PIV measurement results. Despite the usefulness of information were obtained by examining the measured instantaneous flow fields and the timemean 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 turbulent flow data. This technique allows to track turbulent structures in terms of time and scale and extracts new information on turbulence structures (Li, ; Li et al., ) (), (). To extract the instantaneous multiscale turbulent structures from the two-dimensional

2 PIV measurement results in a lobed jet mixing flow, Li et al. () () developed a new signal processing technique, i.e. wavelet vector multi-resolution analysis. For the highly three-dimensional flow fields like lobed jet mixing flows, however, the analysis of two-dimensional measurement results may not be able to reveal its three-dimensional feature successfully. The aim of this paper is to apply the wavelet vector multi-resolution technique to analyze the threedimensional measurement results of a high-resolution stereoscopic PIV system for revealing the instantaneous three-dimensional multi-scale turbulent structures in the near field of lobed jet mixing flow. WAVELET VECTOR MULTI-RESOLUTION TECNIQUE For a two-dimensional vector field f ( x, x ) wavelet basis ( x x ) and a Ψ m, ;,, n m n the two-dimensional discrete wavelet transform is defined by i j ( x, x ) L = i j Wf m n m n f x x Ψ, (, ), ; m, n; m, n i j () The reconstruction of the original vector field can be achieved by f ( x, x ) = Wf Ψ m m n n m, n ; m, n m, n; m, n ( x, x ) () The two-dimensional wavelet basis, Ψ m, ;, (, ) n x x m n, is simply to take the tensor product functions generated by two one-dimensional bases as They provide excellent localization properties both in physical space and frequency space. In this study we use the Daubechies basis with index N=, which is not only orthonormal, but also have smoothness and compact support, to analyze the flow image. The procedure of the wavelet vector multiresolution analysis can be summarized in two steps: () Compute the wavelet coefficients of vector data based on the discrete wavelet transform of Eq. (). () Inverse wavelet transform of Eq. () is applied to wavelet coefficients at each wavelet level, and vector components are determined at each level or scale. The wavelet vector multi-resolution analysis may process fewer data by selecting the relevant details that are necessary to perform an extraction of the multi-scale structures, and decompose the vector data in both Fourier and physical spaces. The technique is unique in terms of its capability to separate turbulence structures of different scales. EXPERIMENTAL SET-UP AND STEREOSCOPIC PIV SYSTEM A test lobed nozzle with six lobes, as shown in Fig., is used in the present study. The width of each lobe is mm and the height of each lobe is mm ( = mm). The inner and outer penetration angles of the lobed structures are θ in = andθ out = respectively. The equivalent nozzle diameter is designed to be D = mm. The z-axis is taken as the direction of the main stream; the x-y plane is perpendicular to the z-axis and is taken as the cross plane of the lobed jet. u, v and w are defined as the velocity components in x, y and z directions, respectively. Figure shows the air jet experimental set-up used in the present study. A centrifugal compressor was used to supply air jet flows. A cylindrical plenum Ψ m, n ; m, n = ( x, x ) ( m + m ) m ( m ψ x n ) ψ ( x n ). () Lobe trough X Lobe peak Lobeside The oldest example of a function ψ ( x) for which the ψ ( x) constitutes an orthogonal basis is the aar m,n 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.. Y Fig. The test lobed nozzle Lobe height =

3 Centrifugal compressor Test nozzle Cylindrical Convergent connection plenum chamber Two-dimensional translation mechanism Fig. The air jet experimental setup optics ost computer Laser sheet Double-pulsed Nd:YAG Laser mm Synchronizer Lobed nozzle Measurement region mm by mm mm high-resolution CCD cameras Fig. The schematic of the stereoscopic PIV system chamber with honeycomb structures was used to settle the airflow. Through a convergent connection (convergent ratio is about :), the airflow is exhausted from the test nozzles. The velocity of the air jet exhausting from the test nozzle can be adjusted and the core jet velocity (U ) was set at about m/s in the present study. The Reynolds number of the jet flow is about, based on the equivalent nozzle diameter (D) and the core jet velocity. Figure shows the schematic of the stereoscopic PIV system used in the present study. The objective jet mixing flows were illuminated by a double-pulsed Nd:YAG laser set (New Wave mj/pulse) with the laser sheet thickness being about mm. The doublepulsed Nd:YAG laser set can supply the pulsed laser at the frequency of z. The time interval between the two-pulsed illuminations was settled as µ s. Two high-resolution (K by K) cross-correlation CCD cameras (TSI PIVCAM-) were used to do stereoscopic PIV image recording. The two CCD cameras were arranged in an angular displacement configuration to get a big overlapped view. In order to have the measurement field focused on the image planes perfectly, tilt-axis mounts were installed between the camera bodies and lenses, the lenses and camera bodies were adjusted to satisfy the scheimpflug condition. In the present study, the

4 distance between the illuminating laser sheet and image recording plane of the CCD camera is about mm, and the angle between the view axes of the two cameras is about. For such arrangement, the size of the overlapped view of the two image recording cameras for stereoscopic PIV measurement is about mm by mm. The two-dimensional particle image displacements in every image planes were calculated separately by using ierarchical Recursive PIV (R-PIV) software () (u et al. ). The ierarchical Recursive PIV software is based on a hierarchical recursive process of conventional spatial correlation operation with offsetting of the displacement estimated by the former iteration step and hierarchical reduction of the interrogation window size and search distance in the next iteration step. RESULTS AND DISCUSSION Instantaneous Three-Dimensional Multi-Scale Velocity Fields In order to gain insight into the multi-scale flow structures, the wavelet vector multi-resolution analysis is applied to the three-dimensional measurement results of PIV. In the present study, the measured three velocity components of x are used. The instantaneous velocity vector u ( x, y) is first decomposed into three wavelet levels and three velocity vector compositions u i ( x, y) within different scale ranges are produced based on the wavelet vector multi-resolution analysis. The velocity vector composition of wavelet level, which corresponds to the central scale of mm, is employed to describe the large-scale flow structure. Then the sum of velocity vector compositions of wavelet levels and, which corresponds to the central scale range of - mm, constructs the smaller flow structure. Of course, the measured velocity vector u ( x, y) can be written as the u i x, y, i.e. sum of velocity vector compositions ( ) u ( x, y) = u ( x, y) = i i, () Figure (a) shows an instantaneous velocity vectors of the stereoscopic PIV measurement in the cross plane ((x, y)-plane view) overlapping on the corresponding the contours of w velocity component at the downstream location of z/d =.. The falsecolors have been assigned to the value of w velocity Fig. (a) The instantaneous velocity field of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (b) The instantaneous velocity field at the central scale of mm in the cross plane of z/d = Fig. (c) The instantaneous velocity field at the central scale of - mm in the cross plane of z/d =.

5 component, and the highest concentration is displayed as read and the lowest as a blue. This is the original data before the wavelet decomposition. The irregular flow structures that imply a multi-scale structure can be observed. The irregular large-scale streamwise vortices can be seen to be in the same cofiguration as the trailing edge geometry of the lobed nozzle. The highest velocity region of w velocity component exists in the center region of the jet flow. The analysis results of the instantaneous velocity vectors of the stereoscopic PIV measurement (Fig.(a)) based on the wavelet vector multi-resolution technique are shown in Fig. (b) and (c), in which the two different scale componets of instantaneous velocity field can be seen. Figure (b) displays the large-scale structures with a central scale of mm. Six pairs of large-scale streamwise vortices can be clearly seen around the edge position of the lobed nozzle, although this location is the initial region of the largescale streamwise vortices generated by the lobed nozzle. These vortices corresponded quite well to the irregular vortices appeared in Fig. (a). This agreement provides a validation for the present data analysis technique. The flow structures with a central scale range of - mm can be shown in Fig. (c). A number of the smaller-scale vortices appeared around the edge position of the lobed nozzle. By comparing the Fig. (b), it is found that these each smaller-scale vortices are contained in the large-scale streamwise vortices. The higher w velocity component within the central scale range of - mm is also identified clearly at the same position. Note that such structure cannot be extracted by traditional techniques. Figure (a) displays the stereoscopic PIV measurement results of the cross plane at the z/d =. The geometry of the lobed nozzle can be identified from the instantaneous velocity vector field. The instantaneous velocity field is found to become more complex than that in the upstream cross plane of z/d =.. The core jet flow is found to diffuse to the ambient flow substantially, and the size of the higher velocity region in the center of the jet flow becomes smaller compared with Fig. (a). The large-scale structure with a central scale of mm, as shown in Fig. (b), can be extracted based on the wavelet vector multiresolution analysis. Several pairs of large-scale streamwise vortices can be clearly seen at the position of lobe. The vector plot of the cross stream flow shows that the streamwise vortices have spread outward. Figure (c) shows the flow structures with a central scale range of - mm. The smaller-scale vortices almost distribut in the whole measured field. Espeically, they are much more active at the position of the trailing edge geometry of the lobed nozzle Fig. (a) The instantaneous velocity field of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (b) The instantaneous velocity field at the central scale of mm in the cross plane of z/d = Fig. (c) The instantaneous velocity field at the central scale of - mm in the cross plane of z/d =

6 Some of them exist in the large-scale streamwise vortices and others exist independently. The region of higher w velocity component is found to be the same cofiguration as the trailing edge geometry of the lobed nozzle. At further downstream location of z/d =, the stereoscopic PIV measurement results are shown in Fig. (a). The geometry of the lobed nozzle almost cannot be identified from the instantaneous velocity field. The region and magnitude of the higher w velocity component in the center of the jet flow are found to decrease rapidly due to the intensive mixing of the core jet flow with ambient flow. Figure (b) gives a clear picture of the large-scale structure with a central scale of mm. The large-scale streamwise vortices can be observed in the range of the equivalent circular nozzle diameter. From the flow structures with a central scale range of - mm, as shown in Fig. (c), many active smaller-scale vortices and the region of the higher w velocity component are identified in the center region of the jet. Instantaneous Multi-Scale Streamwise The above velocity vector plots have shown the existence of very strong multi-scale cross-stream in the lobed jet mixing. The core jet flow expends outward along the lobes and ambient flow ejects inward in the lobe troughs, which result in the generation of large-scale streamwise vortices. Therefore, a pair of counter rotating streamwise vortices in the lobed jet mixing flow can be generated for each lobe. In order to study the evolution of multi-scale streamwise vortices quantitatively, the instantaneous streamwise vorticity was calculated based on the velocity data obtained by the stereoscopic PIV measurement and the wavelet vector multi-resolution analysis. The normalized instantaneous component of streamwise vorticity, ω zi, at scale i can be defined in terms of the derivatives of the instantaneous velocity components viz. D vi ui ω zi = ( ), () U x y where i stands for the scale. Figure (a) shows the distribution of the measured instantaneous streamwise vorticity at z/d =.. The false-colors have been assigned to the vorticity values, and the highest concentration is displayed as red and the lowest as blue, and the positive and negative vorticities are simultaneously denoted by solid and dashed lines, respectively. The alternative positive and Fig. (a) The instantaneous velocity field of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (b) The instantaneous velocity field at the central scale of mm in the cross plane of z/d = Fig. (c) The instantaneous velocity field at the central scale of - mm in the cross plane of z/d =.

7 Fig. (a) The instantaneous streamwise vorticity distributions of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (a) The instantaneous streamwise vorticity distributions of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (b) The instantaneous streamwise vorticity distributions at the central scale of mm in the cross plane of z/d = X mm Fig. (b) The instantaneous streamwise vorticity distributions at the central scale of mm in the cross plane of z/d = Fig. (c) The instantaneous streamwise vorticity distributions at the central scale of - mm in the cross plane of z/d =. negative peaks can be clearly seen around the edge X mm Fig. (c) The instantaneous streamwise vorticity distributions at the central scale of - mm in the cross plane of z/d =

8 negative peaks can be clearly seen around the edge position of the equivalent circular nozzle diameter, which indicates pairs of streamwise vortices. But it is difficulty to identify the smaller-scale structures using the measured instantaneous vorticity. Figure (b) and (c) provides information on the distribution of multiscale vortices in the lobed mixing turbulent jet. The pairs of large-scale streamwise vortices that correspond to vortices appeared in Fig. (a) can be clearly observed in Fig. (b). Figure (c) shows the distribution of the smaller-scale vorticity with a central scale range of - mm. The alternative positive and negative peaks can be clearly seen around the position of lobe, which indicates pairs of the smaller-scale streamwise vortices. It maybe implies the existence of horseshoe vortical structures. As increasing the downstream distance to z/d =, as shown in Fig. (a), the distribution of the measured instantaneous streamwise vorticity exhibits many alternative positive and negative peaks at the position of lobe. They imply the multi-scale pairs of streamwise vortices. owever, Figure (b) only displays the distribution of large-scale streamwise vorticity. Several pairs of large-scale streamwise vortices can be clearly seen at the position of lobe. The distribution of the smaller-scale vorticity with a central scale range of - mm is shown in Fig. (c). As indicated in the above velocity vector plot, many positive and negative peaks that imply the smallerscale vortices appear in the whole measured field. At further downstream location of z/d =, from the distribution of the measured instantaneous streamwise vorticity in Fig. (a), many positive and negative peaks mainly distribute in the center region of the jet. From the results of the wavelet multi-resolution analysis, as shown in Fig. (b) and (c), it is found that both the large-scale streamwise vortices and smallerscale streamwise vortices are concentrated in the center region of jet. The maximum vorticity value of these streamwise vortices is found be decreased when compared with that at z/d =. From the above discuss at different downstream cross planes, it can be seen that as increasing the downstream distance, the size and strength of the large-scale streamwise vortices generated by the lobed nozzle first grow up and appear at the position of lobe. Then they decay rapidly and only appear in the center region of jet. CONCLUSIONS In order to extract the three-dimensional multi-scale structures features of the lobed jet mixing flow, the Fig. (a) The instantaneous streamwise vorticity distributions of the stereoscopic PIV measurement results in the cross plane of z/d = Fig. (b) The instantaneous streamwise vorticity distributions at the central scale of mm in the cross plane of z/d = Fig. (c) The instantaneous streamwise vorticity distributions at the central scale of - mm in the cross plane of z/d =.

9 wavelet vector multi-resolution technique was applied to analyze the three-dimensional measurement results of a high-resolution stereoscopic PIV system in this paper. The following main results are summarized. () The instantaneous three-dimensional flow structures were successfully decomposed into largeand small-scale structures based on the wavelet vector multi-resolution analysis. () The pairs of large- and small-scale streamwise vortices and the higher values of the small-scale w velocity component were found around the position of lobe at the location of z/d=.. () The pairs of large-scale streamwise vortices can be clearly observed at the position of lobe and the small-scale vortices appear in the whole measured field when increasing the downstream distance to z/d =. The higher values of the small-scale w velocity component distribute around the trailing edge of the lobed nozzle. () The large- and small-scale streamwise vortices and the higher small-scale w velocity component appear in the center region of jet at further downstream location of z/d =. REFERENCES () u,., Saga, T. and Kobayashi, T., Research on the Vortical and Turbulent Structures in the Lobed Jet Flow by Using LIF and PIV, Measurement Science and Technology, Vol. (), pp.-. () u,., Saga, T., Kobayashi, T. and Taniguchi, N., Stereoscopic PIV Measurement of a Lobed Jet Mixing Flow, Developments in Laser Techniques and Applications to Fluid Mechanics, R. J. Adrian et al. (Eds.), Springer-Verlag (). () Li., Identification of Coherent Structure in Turbulent Shear Flow with Wavelet Correlation Analysis, ASME Journal of Fluids Engineering, Vol. (), No., pp.-. () Li., Takei, M., Ochi, M., Saito, Y. and orii, 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. (), No., pp.-. () Li., u,., Saga, T., Kobayashi, T. and Taniguchi, N., Extraction of Multi-scale Turbulent Structure from PIV Results based on Wavelet Vector Multiresolution Technique, Proceedings of the th International Symposium on Flow Visualization, UK (), No., pp.-. () u,., Saga, T., Kobayashi, T., Taniguchi, N. and Segawa S., The Spatial Resolution Improvement of PIV Result by Using ierarchical Recursive Operation, Journal of Visualization, No. Vol. ().