Wavelet Multi-Resolution Analysis of Dual-plane Stereoscopic PIV Measurement Results in a Lobed Jet

Transcription

1 th International Symposium on Particle Image Velocimetry Göttingen, Germany, September 7-9, Wavelet Multi-Resolution Analysis of Dual-plane Stereoscopic PIV Measurement Results in a Lobed Jet H. Li, H. Hu, T. Kobayashi, T. Saga, N. Taniguchi Abstract The vector wavelet multi-resolution technique was apply to analyzing the three-dimensional measurement results of a high-resolution dual-plane stereoscopic PIV system for revealing a fundamental understanding of the multi-scale vortical structures in the near field of turbulent lobed jet. The instantaneous threedimensional velocity vectors were decomposed into large- and small-scale velocity fields. The instantaneous pairs of large-scale streamwise vortices can be clearly observed around the edge position of the lobed nozzle and spread outward along the lobes. The stronger small-scale streamwise vortices were found in the lobe regions, which first spread outward along the lobes and then develop to the whole measured flow field. These small-scale streamwise vortices also play an important role in enhancing mixing process. It is also found that the strength of the large-scale azimuthal Kelvin-Helmholtz vortices is higher than that of the streamwise vortices. At the downstream the stronger large-scale azimuthal vortices are concentrated in the region of jet core. 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 more efficiently. The distortion of azimuthal vortex structure may lead to streamwise vortices under certain conditions. 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 large-scale streamwise vortices and to induce rapid mixing, which has been considered to be one promising method for the enhanced jet mixing process. More recently, it has become obvious that lobed mixers provide the most mixing enhancement in the presence of a velocity difference or normal vorticity component. The first application of the lobed mixer was for jet noise reduction and net thrust increase of a turbofan jet engine (Crouch et al., 977), because of the mixing characteristics of lobed mixers and their advantage over the convectional nozzle designs. Now the lobed mixer has been applied to the combustion chambers in engine, the spread of pollutants at industrial sites and so on, and becomes an extraordinary fluid mechanic device for efficiently mixing two different flow streams in determining the length. Due to these practical applications in engineering, the investigation of the detailed mixing mechanisms becomes significant. Paterson (98) provided the first detailed experimental data in the mixing region downstream of axisymmetric lobed nozzles by using laser Doppler velocimetry (LDV). Originally, it was believed that the mixing enhancement mechanism was solely due to the increased interfacial area. However, his studies revealed that the flow field of downstream was dominated by strong secondary flow structures along with relatively large-scale streamwise vortices having scales on the order of the lobe height, which played a major role in the enhanced mixing process. Also, a horseshoe vortex on the order of the lobe half-height was found to exist in the lobe troughs. The contribution of these vortices to the overall mixing process of each was not clear, but it was believed that these vortices to have been created by the lobe geometry and the trailing edge Kutta condition. Most of the later studies concentrated on discovering the underlying physics of the twodimensional lobed mixing process. Werle et al. (987) found that the vortex formation process was inviscid effect and the mixing process takes place in three basic steps: the vortices form, intensify, and then rapidly break down H. LI, Kagoshima University, Kagoshima, Japan H. Hu, Michigan State University, Michigan, USA T. Kobayashi, T. Saga, N. Taniguchi, University of Tokyo, Tokyo, Japan Correspondence to: Dr. H. LI, Department of Mechanical Engineering, Kagoshima University, --, Korimoto, Kagoshima 89-, Japan, li@mech.kagoshima-u.ac.jp

2 into small-scale turbulence. Eckerle et al. (99) studied mixing downstream of a lobed mixer at two velocity ratios using a two-component LDV. It was found that the breakdown of the large-scale vortices and the accompanying increase in turbulent mixing are significant parts of the mixing process. Ukeiley et al. (99, 99) applied the proper orthogonal decomposition (POD) to a rake of single-component hot-wire data obtained in a lobed mixer flowfield. They showed that the large-scale vortex, as defined by the POD, breaks down, and the flow becomes more homogeneous. McCormik (99) reported more detailed experimental investigation of the vortical and turbulent structure using the pulsed-laser sheet flow visualization with smoke and three-dimensional velocity measurements taken with a hot-wire. Their study confirmed a new vortex structure existing in addition to the wellknown streamwise vortex array, and found that the syreamwise vorticity deforms the normal vortex into a pinchedoff structure that may also enhance small-scale turbulent mixing. Recently, the mixing process in a coaxial jet where the inner nozzle is a lobed mixer was experimentally studied by Belovich et al. (99, 997). Detailed flow visualization and velocity measurements with LDV were performed, and different mixing mechanisms for each velocity ratio were discussed. 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. Hu et al. (a, a,b) employed two-dimensional, stereoscopic and dualplane 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. Despite the usefulness of information were obtained by examining the measured instantaneous flow fields and the timemean turbulent quantities, further detail of the mixing process associated with 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. The continuous wavelet transform has been proposed to analyze turbulent structures in terms of time and scale by Li (99, 998). The coefficients of the continuous wavelet transform are known to extract the characterization of local regularity continuously. However, it is unable to reconstruct the original function because the mother wavelet function is non-orthogonal function. It is important to reconstruct the original signal from wavelet composition in studying multi-scale turbulent structures. On the other hand, the discrete wavelet transform allows an orthogonal projection on a minimal number of independent modes and is invertible. Such analysis can produce a multi-resolution representation and might be also used to analyze turbulent flows. Charles () first used the one-dimensional discrete wavelet transform to obtain local energy spectra and the flux of kinetic energy from experimental and direct numerical simulation data. Li et al. (999, b) employed discrete wavelet transforms to evaluate eddy structures of a jet in the dimension of time and scale. Li et al. (999, b) also applied the twodimensional orthogonal wavelets to turbulent images, and extracted the multi-resolution turbulent structures. For decomposing turbulent flows into coherent and incoherent structures, Farge et al. (999) proposed a coherent vortex simulation method based on orthogonal wavelets. To extract multi-scale turbulent structures of a lobed jet mixing flow, Li et al. (, ) developed vector wavelet multi-resolution technique to analyzing the two-dimensional and stereoscopic PIV measurement results. The purpose of this paper is to apply vector wavelet multi-resolution technique to analyze the three-dimensional measurement results of a high-resolution dual-plane stereoscopic PIV system in the near field for providing a fundamental understanding of the multi-scale vortical structures and why vorticity dynamics greatly impact the mixing process of lobed jet. Vector Wavelet Multi-Resolution Technique Here, we consider analyzing a two-dimensional vector field f ( x, x ). The simplest way of constructing a twodimensional orthogonal wavelet basis is to take the simple production of two one-dimensional orthogonal wavelet bases: Ψ m m m ( x, x ) = ( x n ) ψ ( x n ) m ; n, n ψ, () where m denote the dilation index and n and n represent the translation index. 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=, which is not only orthonormal, but also have smoothness and compact support, to analyze the vector field.

3 The two-dimensional discrete vector wavelet transform is defined by i j ( x x ) = i j Wf m; n n f ( x, x ) Ψ, m; n, n,. () i j As in the Fourier series, dilation by larger m compresses the vector field on the x- and y-axis. Altering n and n has the effect of sliding the vector along the x- and y-axis, respectively. The reconstruction of the original vector field can be achieved by f ( x, x ) = Wf Ψ ( x x ) m; n n m; n, n,, m n n. () The goal of the wavelet multi-resolution analysis is to get a representation of a signal or an image that is written in a parsimonious manner as a sum of its essential components. That is, a parsimonious representation of a signal or an image will preserve the interesting features of the original signal or image, but will express the signal or image in terms of a relatively small set of coefficients. It is well known that vector field often includes too much information for multi-scale vision processing. Multi-resolution algorithm can process fewer data by selecting the relevant details that are necessary to perform a particular recognition task. Coarse to fine searches processes first a low-resolution vector field and zoom selectively into finer scales information. In this study, the procedure of the wavelet vector multi-resolution 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. Of course, a sum of these essential vector components can recover the original vector field. The vector wavelet multi-resolution analysis can perform an extraction of the multi-scale structures, and decompose the vector field 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 Dual-plane 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 (H = 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. 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 dual-plane stereoscopic PIV system used in the present study. The objective jet mixing flows were illuminated by two sets of widely used double-plused Nd:YAG lasers (New Wave, mj/pulse) with the laser sheet thickness being about mm. The gap between the centers of the two illuminated planes is adjusted as. mm. The additional optics (half wave plate, mirrors, polarizer and cylinder lens) was also inner penetration angle X Lobe peak outer penetration angle Lobe trough Y Lobe height H=mm Fig. The test lobed nozzle Z

4 Mirror # S-polarized laser beam cylinder lens Mirror # P-polarized laser beam Half wave (λ/) plate Double-pulsed Nd:YAG Laser set A Double-pulsed Host computer Laser sheet with S-polarization direction Lobed nozzle Polarizer cube Laser sheet with P-polarization direction Measurement region 8mm by 8mm Polarizing beam splitter cubes Nd:YAG Laser set B Mirror # high-resolution CCD camera mm Synchronizer Mirror # mm high-resolution CCD camera high-resolution CCD camera high-resolution CCD camera Fig. The schematic of the dual-plane stereoscopic PIV system used to set up the illumination system of the dual-plane stereoscopic PIV system. The double-pulsed Nd:YAG laser set can supply the pulsed laser at the frequency of Hz. The time interval between the two-pulsed illuminations was settled as µ s. Two pairs of high-resolution cross-correlation CCD cameras (K by K, 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 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 system is about 8 mm by 8 mm. The Hierarchical Recursive PIV (HR-PIV) (Hu et al. ) method was used to calculate the two-dimensional particle image displacements in each image plane. The window of pixels was employed for the final step of the recursive correlation operation. The spatial resolution of the present stereoscopic PIV measurement is about mmxmmxmm. Finally, by using the mapping functions obtained by the in-situ calibration and the twodimensional displacements in the two image planes, all three components of the velocity vectors in the illuminating laser sheet plane were reconstructed. More details of the dual-plane stereoscopic PIV system have been given in Hu et al. (b). Results and Discussion. Instantaneous Three-Dimensional Multi-Scale Velocity Fields in Dual-plane As described in previous studies, two kinds of structures are very important for the mixing process in lobed mixing flows: One is the large-scale streamwise vortices generated by the special trailing edge of the lobed nozzle, another is the zaimuthal vortices rolled up at the interface of the shear layers due to the Kelvin-Helmholtz instability. However, it is impossible to analyze quantitatively the mixing process of lobed flows at various scales from the measurement results of the present dual-plane stereoscopic PIV due to the limitation of the traditional experimental analysis techniques. For gaining insight into the multi-scale flow structures, the vector wavelet multi-resolution analysis is applied to the three-dimensional measurement results of dual-plane stereoscopic 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 compositions, u i ( x, y), with different wavelet levels or scales, where i represents the scale. Then three velocity vector compositions, u i ( x, y), within different scale ranges are produced based on vector wavelet multi-resolution analysis. The velocity vector composition of wavelet level, which corresponds to the

5 central scale of mm, is employed to describe the large-scale flow structure. The sum of velocity vector compositions of wavelet levels and, which corresponds to the central scale of mm, constructs the smaller scale flow structure. Of course, the measured velocity vector u ( x, y) can be written as the sum of three velocity vector u i x, y, i.e. compositions ( ) u ( x, y) = u ( x, y) = i i. () Since the characteristics of the mixing process in the lobed jet mixing flow revealed from the velocity distributions has been discussed extensively in the earlier works, the majority of the results and discussions given in the present paper will focus on the characteristics of the turbulent structures of various scales in the instantaneous dual-plane. Figure shows a pair of instantaneous large-scale velocity field of the dual-plane stereoscopic PIV measurement (in the cross plane ((x, y)-plane view) overlapping on the corresponding the contour of the axial velocity component in two parallel cross planes (Z = 8 and 8 mm) with a central scale range of mm. The false-colors have been assigned to the value of w velocity component, and the highest concentration is displayed as read and the lowest as a blue. The original measurement data and their discussion before the wavelet decomposition have been found in Hu et al. (b). In this paper we only analyze the velocity fields of various scales. From distributions of the instantaneous velocity vector field and contour of axial velocity, the geometry of the lobed nozzle can also be identified. The vector plot clearly reveals that the steamwise vortices or large-scale structures appear around the edge position of the lobed nozzle and spread outward along the lobes. At further downstream locations of Z = and mm, the instantaneous large-scale velocity fields with a central scale of mm are shown in Fig.. The geometry of the lobed nozzle almost cannot be identified. The largescale vortices almost disappear and the streamwise vortices can be observed in the almost whole measured flow field. Especially, they are much more active in the jet core region. The higher axial velocity component is found in the center of the jet flow and decreases rapidly due to the intensive mixing of the core jet flow with ambient flow. The instantaneous velocity fields with a central scale of mm in two parallel cross planes of Z = 8 and 8 mm are plotted in Fig.. The small-scale streamwise vortices are much more active at the position of the trailing edge geometry of the lobed nozzle. It is also found that some of these small-scale vortices are contained in the large-scale streamwise vortices. It suggests that the small-scale structures obtain the energy from the large-scale structures and enhance the small-scale mixing process around the lobe regions. Figure gives the velocity field with a central scale of mm at downstream locations of Z = and mm. The velocity vectors indicate that the active streamwise small-scale vortices distribute in the whole measured flow field. The contour of the axial velocity component indicates that the alternate positive and negative peaks rapidly decrease and become weak. It implies that the spanwise vortex ring has beak down into many disconnected vortical tubes (a) Z = 8 mm (b) Z = 8 mm Fig. The instantaneous velocity field with the central scale of mm in the cross plane of Z = 8 and 8 mm

6 (a) Z = mm (a) Z = mm Fig. The instantaneous velocity field with the central scale of mm in the cross plane of Z = and mm (a) Z = 8 mm (a) Z = 8 mm Fig. The instantaneous velocity field with the central scale of mm in the cross plane of Z = 8 and 8 mm (a) Z = mm (b) Z = mm Fig. The instantaneous velocity field with the central scale of mm in the cross plane of Z = and mm

7 . Instantaneous Multi-Scale Streamwise in Dual-plane The three normalized vorticity components at scale i in the lobed jet mixing flow may be definted by defined in terms of the derivatives of the instantaneous velocity components viz. D w v ( i ϖ i xi = ), U y z () D u w ( i ϖ i yi = ), U z x () D vi ui ϖ zi = ( ), U x (7) y where D is the diameter of the lobed nozzle, and U is the exit jet velocity. While, u i, v i and w i are the instantaneous velocity componets at scale i in X, Y and Z directions (Fig.). Since the velocity fields at two illuminated planes can be simultaneously measured by the present dual-plane stereoscopic PIV system, it is possible that all vorticity components at each scale i can be calculated after wavelet multi-resolution analysis. Besides the out-of-plane component componet ϖ zi, the other two in-plane components of the vorticity ( ϖ xi and ϖ yi ) can also be obtained in the two illuminated planes with first-order approximation accuracy. In order to reveal the azimuthal vortices rolled up in the lobed mixing flows due to the Kelvin-Helmholtz instability, the x- and y-components of the vorticity at scale i were combined into in-plane vorticity by in plane, i = ϖ xi ϖ yi. (8) ϖ + Although the existences of the streamwise vortices and the azimuthal Kelvin-Helmholtz vortices in lobed mixing flow have been confirmed in previous studies, the quantitative analysis about the distributions of the streamwise and azimulthal vorticity at various scales is not clear. The wavelet vector multi-resolution analysis of dual-plane stereoscopic PIV results make it possible to reveal instantaneously the streamwise vortices and the azimuthal Kelvin-Helmholtz vortices. Since the evolution of the large- and small-scale streamwise vortices has been studied in our previous paper (Li et al. a), we place emphasis on discussing the large- and small-scale structures of the azimuthal Kelvin-Helmholtz vortices and on making a comparison between the azimuthal Kelvin-Helmholtz and streamwise vortices this paper. Contours of the instantaneous large-scale vorticity with a central scale of mm at Z = mm is shown in Fig. 7. The B&W false-colors have been assigned to the vorticity values. The highest concentration is displayed as black and the lowest as white, and the positive and negative vorticity are simultaneously denoted by solid and dashed lines, respectively. As shown in Fig. 7(a), the alternative positive and negative peaks can be seen around the lobe edge positions, which indicate the pairs of streamwise vortices. However, the distributions of other two vorticity components ( ϖ xi and ϖ yi ) in Fig.7 (b) and (c) exhibits clearly the geometry of the lobed nozzle. The stronger positive and negative peaks can be clearly seen around the lobe positions, which indicate the large-scale azimuthal Kelvin-Helmholtz vortices. It is also found that the strength of the large-scale streamwise vortices is weaker than that of the other vorticity components. The distribution of the azimutha (in-plane) vorticity is shown in Fig.7 (d). It is evident that the large-scale azimuthal Kelvin-Helmholtz vortices distribute around the lobe positions. The strength of the large-scale azimuthal Kelvin-Helmholtz vortices is higher than that of the streamwise vortices. In the case of the small-scale, as shown in Fig.8, the positive and negative peaks of three vorticity components can be clearly observed at the trailing edge of the lobed nozzle. The strength of the small-scale azimuthal Kelvin-Helmholtz vortices is lower than that of the large-scale azimuthal Kelvin-Helmholtz vortices, but the small-scale streamwise vortices exhibit higher strength than that of the large-scale streamwise vortices. It implies that the large-scale streamwise vortices decay and breakup rapidly along flow direction, but the large-scale azimuthal Kelvin-Helmholtz vortices develop and dominate the flow structures. At a downstream distance of Z = mm, as shown in Figs.9 and, many positive and negative peaks distribute in the all flow region. Especially the small-scale vortices with a central scale of mm appear in the whole measured field. The geometry of the lobed nozzle cannot be identified from the distribution of vorticity. The strength of largeand small-scale vortices decreases comparing to Figs.7 and 8, and the magnitude of the small-scale vorticity becomes larger than that of the large-scale vorticity. It also found that the azimuthal vorticity still remains the higher 7

8 - - - (Z-component) (X-component) (a) Z-component vorticity (b) X-component vorticity (Y-component) In-plane (c) Y-component vorticity (d) in-plan vorticity Fig. 7 The instantaneous vorticity distributions with the central scale of mm in the cross plane of Z = mm (Z-component) (X-component) (a) Z-component vorticity (b) X-component vorticity 8

9 - - - (Y-component) In-plane (c) Y-component vorticity (d) in-plan vorticity Fig. 8 The instantaneous vorticity distributions with the central scale of mm in the cross plane of Z = mm - (Z-component) (X-component) (a) Z-component vorticity (b) X-component vorticity - (Y-component) In-plane (c) Y-component vorticity (d) in-plan vorticity Fig. 9 The instantaneous vorticity distributions with the central scale of mm in the cross plane of Z = mm 9

10 - (Z-component) (X-component) (a) Z-component vorticity (b) X-component vorticity - (Y-component) In-plane (c) Y-component vorticity (d) in-plan vorticity Fig. The instantaneous vorticity distributions with the central scale of mm in the cross plane of Z = mm value than the streamwise vorticity for both of the large- and small-scale. From the instantaneous azimuthal vorticity given in Fig.8 (b) and (c), it can seen that the stronger large-scale azimuthal vorticity are concentrated in the region of jet core. Conclusions In order to reveal the three-dimensional multi-scale structure features of the lobed jet mixing flow, the vector wavelet multi-resolution technique was apply to analyzing the three-dimensional measurement results of a highresolution dual-plane stereoscopic PIV system at two spatially separated planes. The following main results are summarized. () The instantaneous velocity vector field and contour of axial velocity reveals that the large-scale structures appear around the edge position of the lobed nozzle and spread outward along the lobes. () The stronger small-scale streamwise vortices have been confirmed to exist in the lobe regions at Z = 8 and 8 mm. These vortices first become intense and spread outward along the lobes, and then develop to the whole measured flow field after Z = and mm. It indicates that the small-scale streamwise vortices also play an important role in enhancing mixing process. () The strength of the large-scale azimuthal Kelvin-Helmholtz vortices is higher than that of the treamwise vortices. () At the downstream the stronger large-scale azimuthal vortices are concentrated in the region of jet core.

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