A MHZ RATE IMAGING SYSTEM FOR STUDY OF TURBULENT AND TIME EVOLVING HIGH SPEED FLOWS

Transcription

1 A MHZ RATE IMAGING SYSTEM FOR STUDY OF TURBULENT AND TIME EVOLVING HIGH SPEED FLOWS B. Thurow, J. Hileman, M. Samimy, and W. Lempert Department of Mechanical Engineering The Ohio State University Columbus, OH ABSTRACT We present a second generation MHz frame rate imaging system, based on a custom fabricated "pulseburst" laser and a commercial CCD framing camera. The utility of the system for high-speed flow visualization is demonstrated in a Mach 1.3 axisymmetric jet. Complete 17 frame image sequences are used to explore the evolution and interaction of large scale structures and to determine their convective velocity, employing a novel spatial correlation technique, which includes a "translating template" that convects downstream with the structure being tracked. Preliminary progress towards the development of a MHz rate Planar Doppler Velocimetry diagnostic is also included. INTRODUCTION The last decade has seen a literal explosion in the development and application of new laser-based measurement techniques for probing dynamics of fluid flows. The ability to capture time-evolving or volumetric information, however, is severely constrained by limitations of available technology. As a general rule, in the gas phase, signal levels for demonstrated imaging diagnostic techniques are sufficiently small that the measurements require high power, nanosecond duration, pulsed sources, such as Q- switched solid-state (principally Nd:YAG) or excimer lasers. The pulse repetition rate of these devices is limited to the range pulses/second for solid-state lasers, and pulses/second for excimer lasers. Typical commercially available, high sensitivity digital image sensors, such as CCD cameras, are similarly constrained. This is, in general, too slow, by many orders of magnitude, to track dynamical behavior in high speed - high Reynolds number flow. In this paper, we describe the design and performance of a MHz rate flow imaging system, and demonstrate its utility as a high speed flow diagnostic tool, with a particular emphasis on flow visualization measurements in a Mach 1.3 axisymmetric free jet. The use of the system for determination of convective velocity, as well as recent progress towards development of Real Time Planar Doppler Velocimetry (RTPDV), will also be presented. FACILITIES AND INSTRUMENTATION Laser System The laser, illustrated in Fig. 1, is a second generation system based on that described previously by Lempert et al. 1, Wu, et al. 2 and Thurow et al. 3,4 A continuous wave Nd:YAG ring laser serves as the primary oscillator, the output of which is pre-amplified in a double-pass flashlamp-pumped, pulsed amplifier. The resulting, approximately 150 microsecond duration pulse is formed into a "burst" train using a custom, dual Pockel Cell "slicer". The train can have a variable number of pulses, between 1 and 99, with interpulse spacing varying between 1 and 100 microseconds, and individual pulse duration variable between 5 nsec and 10 microseconds. For the experiments to be presented, a typical burst consisted of 17 pulses (constrained by the imaging camera), with 10 nsec duration and 2-10 microsecond spacing. The pulse train is further amplified by two double-pass and one single pass amplification stages, and then converted to the second harmonic wavelength of microns ("green") using a KTP crystal. CW LASER (1064 nm) MIRROR WAVE PLATE λ/2 FOCUSING LENSE OPTICAL ISOLATOR POLARIZER HARMONIC CRYSTAL TO APPLICATION (532 nm) FAST POCKEL S CELLS λ/2 λ/4 λ/4 λ/4 λ/2 λ/2 BEAM EXPANDING LENSES AMPLIFIER Figure 1: Schematic Diagram of Pulse Burst Laser

2 Figure 2 shows a typical burst train, consisting of 17 pulses, with individual pulse duration of 10 nsec and inter-pulse spacing of 5 microseconds. The output energy, at the second harmonic wavelength, is approximately 7 mj in each of the individual pulses comprising the burst. In general, the power in each individual pulse varies between 2 and 15 mj depending, primarily, upon the length of time between the pulses. Longer separations correlate to higher pulse energies. The overall burst process has a repetition rate of 10 Hz. While not specifically examined in this work, previous studies 5 have determined the spectral characteristics of the burst mode laser to be nearly ideal, with measured spectral linewidth (FWHM) of order 50 MHz for a 16 nsec duration square temporal pulse. While not critical for flow visualization, the linewidth and pulse-to-pulse spectral stability of the laser is critical if it is to be employed for PDV. Relative Power Time (microseconds) Figure 2: Typical burst train for MHz Pulse Burst Laser; pulses are separated by 5 microseconds. Power in individual pulse is approximately 7mJ. Camera System The camera used in these experiments is manufactured by Silicon Mountain Design (SMD) (now a subsidiary of Dalsa, Inc.). The camera can acquire 17 images at a variable rate as fast as 1 MHz. Each image in the sequence has a resolution of 240 x 240 pixels. The camera is based on a large format (960 x 960) 12 bit CCD chip in which fifteen of sixteen pixels are hidden by a custom fabricated mask. The individual pixel size is 14 x 14 microns, which, as will be discussed in the last section, significantly impacts Real Time PDV measurements. By appropriate shifting of charge, individual images are initially stored in pixel locations under the mask. After accumulation of all the images, a PC computer reads the output as one large image, containing all 17 frames. The seventeenth frame is obtained by using what is known as the CCD drain, which is normally used to dump dark charge collected in the interval between frames. Mach 1.3 Jet Facility The experiments were conducted at The Ohio State University s Gas Dynamics and Turbulence Laboratory (GDTL). The facility consists of a jet stand and stagnation chamber to which a variety of nozzles may be attached. Air is supplied to the stagnation chamber from two four stage compressors; it is filtered, dried and stored in two cylindrical tanks with a total capacity of 42.5 m 3 at 16.5 MPa (1600 ft 3 at 2500 psi). The stagnation chamber contains a perforated plate and two screens of varying porosity to condition the flow to be as uniform and steady as possible prior to entering the nozzle. The facility has been described in more detail elsewhere 6,7. The jet nozzle used for the majority of work to be presented here has a 1 (25.4 mm) exit diameter and a design Mach number of 1.3, the contour of which was designed using the method of characteristics. The jet exhausts into an anechoic chamber and exits the facility through a large bell-mouth at the opposite end of the chamber. Pressure to the stagnation chamber is controlled manually through the actuation of a Fisher control valve and can be maintained at constant pressure within 0.1 psi. Pressure was set for an ideally expanded flow and held constant through each set of experiments. SUMMARY OF EXPERIMENTS Flow Visualization BACKGROUND We have recently reported 4 a detailed flow visualization study of a Mach 1.3 axisymmetric jet, the results of which will be summarized here. The axisymmetric geometry is particularly interesting because it contains elements of the more commonly studied planar shear layer, but also exhibits additional complexity due to the interaction of the two mixing layers across the jet core. Figure 3 shows a simple schematic of the development of the mixing layer, which consists of two regions, based on structure growth and development. In the first region structures develop without direct interaction from structures on the opposite side. In the second region (anywhere from 5 to 12 jet diameters downstream) structures have grown to sufficient size that they begin to interact with one another across the jet s core. This is designated as a cross-mixing layer interaction. It is worth noting that this second region constitutes the primary source of acoustic noise in jets. A recent study by Hileman et al. using the imaging

3 system employed in this study was conducted to examine the correlation between the noise produced from this region of the flow and the development and interaction of large-scale structures. 8 A common parameter used to quantify the compressibility of a shear flow is the convective Mach number, M c, which represents the convective velocity of structures in the mixing layer. The convective Mach number is theoretically given as M c = U U ) /( a + ) (1) ( a2 Mixing Layer Structure Development Cross-Mixing Layer Interaction Figure 3: Simple schematic of a high-speed axisymmetric flow. where U 1 and U 2 are the velocities of the high-speed and low-speed streams, respectively and a 1 and a 2 are the speeds of sound 9. Many researchers have investigated the effects of compressibility on largescale structures in planar shear layers, base flows, 13 and jets, including efforts using high-speed cinematography. 17,18 It has been found that for relatively low levels of compressibility (M c <0.4) the shear layer generally has a two-dimensional structure with large organized spanwise rollers. As the compressibility level is increased (M c >0.6), large-scale structures become less organized, more threedimensional, and oblique instabilities begin to appear. Of particular relevance to this work are previous reports that, at these higher levels of compressibility, the convective velocity deviates from the prediction of equation (1) according to the stream selection rule. This rule states that for shear flows with supersonic and subsonic streams, large-scale structures will exhibit higher convective velocity than predicted. 9,16 DEVELOPMENT OF STRUCTURES WITHOUT ACROSS THE JET CORE INTERACTION Figure 4 shows a typical example of streamwise images. Seeding is provided via product formation, whereby moisture in the warm ambient air condenses into small 50 nm particles as it mixes with cold dry core fluid. These images provide the most familiar view of structure development as the mixing layer in these images looks very similar to previously published images of planar shear layers at similar convective Mach numbers. The left edge of the image is 4.5 jet diameters downstream while the right edge is at 9 jet diameters. Flow is from left to right and the images are separated in time by 5 microseconds. Very little interaction, if any, is seen to occur across the mixing layer. Note that even though the average end of the jet core is about 6 jet diameters from the nozzle exit, 6,8 the jet core extends far beyond that in many of the movies, example of which is in Fig. 4. The initial formation and evolution of structures is an interesting feature of the flow in this region. For example, numerous movies depict large-scale structure formation by structure roll up. In this mechanism, ambient fluid that penetrates into the high-speed core moves at faster speed than the fluid in the outer mixing layer region. This causes the mixing layer to fold over upon itself, with the forming structure taking on a rotational motion. This development can clearly be seen in the bottom mixing layer in Fig. 4, indicated by the rectangles. It is also clear, however, that large-scale structures in the mixing layer are subjected to large shear forces, which tear and tilt them in the streamwise direction. More specifically, the portion of the structure in the high-speed portion of the flow is stretched forward while the low speed portion appears to be left behind. In the related phenomenon of pairing, adjacent large-scale structures undergo the tilting described above. In this process the high-speed portion of one structure catches up with the low-speed portion of the structure immediately downstream of it. These two portions then begin to rotate about one another, tearing apart the original structures. This has been previously described as fractional pairing by Hussain 19, who provides an excellent schematic of this process. The oblong shapes in Fig. 4 also illustrate this type of pairing as two large structures are initially shown in the process of tilting and then begin to rotate together. It should be pointed out that the above example was specifically chosen to illustrate how the same development in a structure s life might be interpreted in two different ways. The rectangular box was used to illustrate the roll up of a structure, while the rhombuses were used to illustrate the rotational pairing of two structures. These two illustrations, however, are following the same region of the mixing layer and the same event in the mixing layer s growth. This type of occurrence is seen quite often in this region of the mixing layer and serves as one illustration of the added information obtainable from the MHz rate imaging system.

4 4½ x/d 9 x/d Figure 4: Sequence of sixteen streamwise images illustrating structure development in absence of cross core interaction. Time delay between images is 5 microseconds, ordered from top to right, top to bottom. Rectangular boxes illustrate how structure development can be viewed as do to fluid roll-up. The oblong boxes illustrate how the same process can be viewed as due to tilting and pairing of two structures. Figure 5 is a typical example of a sequence of crossstream images located at 3 jet diameters downstream. The images are separated by 4 microseconds and the image is cropped so that it is approximately 1 ¾ wide (and tall). Clearly the mixing layer is not uniform around the edge and the three-dimensional nature of structures can be seen. Furthermore, the development of the inner edge of the mixing layer is very rapid with respect to the time separation of the images. This is an indication of the small scales that are present throughout the layer as well as the high speed of the flow. Another key feature of the flow is the presence of streamwise vorticity. While vorticity cannot be directly extracted from these movies, it can be inferred from the rotational motion of fluid, which, although easier to visualize when viewed as a movie, can be seen at a number of azimuthal locations in Fig. 5. These vortices are believed to be partly responsible for entrainment and ejection. This is most apparent on the outer edge of the mixing layer where mushroom shaped structures are often seen to protrude into the ambient air, thus providing a means for fluid ejection. The circle in the first image points to what appears to be two counterrotating vortices that seem to be entraining ambient fluid into the mixing layer.

5 1¾ Figure 5: Sequence of sixteen cross-stream images obtained at location three jet diameters downstream from exit. Time delay between images is four microseconds, ordered from top left to bottom right. Circles indicate apparent counter-rotating vortex pairs. DEVELOPMENT OF STRUCTURES WITH ACROSS THE JET CORE INTERACTION As the flow progresses downstream, its properties are similar, but typically larger in scale. Structures are also more clearly identifiable and the wavy nature of the mixing layer is more pronounced. The presence of streamwise vorticity is also more easily inferred as larger vortices are now causing fluid to swirl in a more dramatic fashion. Furthermore, as structures continue to grow in size, it is clear that they reach a point where they will begin to influence one another, both from the same and from opposite sides of the mixing layer. Figure 6 is a streamwise sequence of images of one such interaction. In these images, the left edge is about 5⅝ jet diameters downstream and the image extends to approximately 11 jet diameters downstream. Ten microseconds separate each image. The increased organization of structures is clearly evident in the last few images in the sequence. It also appears that these structures formed via a roll up of the mixing layer.

6 5 5/8x/D 11 x/d Figure 6: Sequence of stream wise images illustrating cross core interactions. Time delay between images is 10 microseconds. Boxes illustrate the development of a pair of large structures formed via roll up of the mixing layer. Cross-stream images were obtained at 9 and 12 D downstream and provide for a more complete (and more complex) picture of the jet. As seen in the streamwise movies this is generally the location where interactions between large scale structures begin to take place. Additionally, the cross-stream movies help shed further light on the three-dimensional nature and size of the interacting structures. Figure 7 is a sequence of images separated by 4 microseconds at 9 jet diameters downstream. While the mixing layer still has a semblance to its original circular form, it is apparent that the rapid growth of structures at different azimuthal locations has caused the mixing layer to have a highly asymmetric shape. The rapid dynamics of the jet core is also apparent in this sequence, as is the increased interaction across the jet core. These cross core interactions are also clearly shown to be very three-dimensional. In many cases, two structures on opposite sides of the mixing layer grow and consequently pinch the jet s core. In other cases, it appears that one structure grows significantly larger than any other structure seen in the cross-stream images and the mixing layer is dramatically dominated by this one structure. Additionally, ejection events are much larger in magnitude in this region and cause a high degree of mixing that also modifies the growth of the mixing layer. This is evident in Fig. 7. These events appear to cause an increase in mixing and a consequent growth in the mixing layer. These events, and consequently streamwise vorticity, play a key role in the entrainment characteristics of the jet.

7 2 Figure 7: Sequence of cross stream images obtained at a location 9 jet diameters downstream from exit. Time delay between images is 4 microseconds. Images illustrate rapid evolution of jet core with outer edge dominated by large ejection events. CONVECTIVE VELOCITY MEASUREMENT Background Streamwise mage sequences such as those demonstrated in Figs 4 and 6 are ideal for determination of convective velocity, which is defined as the velocity with which large scale structures convect with the flow. From eqn. (1) the predicted value for this jet is 206m/sec. It has been reported, however, that for convective Mach numbers above 0.4, eqn (1) no longer is valid and the convective velocity is expected to be higher. Murakami and Papamoschou provide an empirical relationship for the deviation from theory for the convective velocity of structures in axisymmetric co-flows. 16 Substituting this experiment s flow conditions into this empirical relation gives a convective velocity of 311 m/sec. In order to compare the experimental results obtained in this work with this empirical relationship, a crosscorrelation scheme was developed to track structures through the movie. The method has been described in detail previously 4 and will, therefore, be only summarized here. The basic idea is to use the twodimensional cross-correlation to identify a structure as it convects downstream in the mixing layer through the course of the movie. This procedure has been employed by several researchers on image pairs obtained using either two lasers or a single double-

8 pulsed laser ,16-18 The point where the crosscorrelation coefficient is the maximum is then taken as the displaced location of the structure. Our approach is similar except that information is used from each of the up to 17 images in each movie sequence. Procedure And Results All correlations employed a template of dimensions 2δ wide x 1.5δ tall, where δ is the average visual thickness of the mixing layer for a given set of movies. (The rectangle in Fig. 4 is roughly the size of the templates used). Prior to correlation, individual images are spatially filtered using a 3 x 3 averaging filter. The initial transverse position of the template was chosen to be the center of the mixing layer on the upper side of the layer. The initial streamwise location of the template was determined by scanning the window across the mixing layer in the first image, choosing the location that contains the highest total intensity. Cross correlation is carried out using a template, which translates with the structure every three to four images. For example, after determination of the spatial displacement in image four, corresponding to peak correlation with image one, the template is shifted by this amount. This new location is then used to determine the correlation between images four and five. The template is then displaced onto another image, and so on. In this fashion, the template is adapted to the structure as it convects with the flow and, more importantly, as it spatially evolves. Note that the template is not translated after every image in order to maintain a minimum translation of order pixels, which was found to improve accuracy. was also set that each of these four calculations must agree to within 10% of their average. An additional criterion was imposed that required the streamwise positions of the templates to all be within a one-half jet diameter of one another, in order to ensure that the procedure is tracking the same structure over the entire image sequence. Figure 8 is a histogram of velocity data obtained from the ~275 image sequences which met these two criteria, incorporating correlations which covered the spanwise range between two and thirteen jet diameters. The mean velocity is 258 m/sec, which is considerably higher than the value of 209 m/sec predicted by eqn (1) and lower than the empirical prediction of 311 m/sec. 16 More discussion on this disagreement and the convective velocity routine can be found in Thurow 20 REAL TIME PDV Finally, we briefly summarize recent progress towards our ultimate goal of developing quantitative Real Time PDV. PDV utilizes the frequency shift of laser light after it is scattered by moving particles. Similar to planar imaging velocimetry (PIV), the technique yields an instantaneous snapshot of the velocity field. More specifically, when a narrow spectral linewidth laser is incident to a flow field, scattering (by particles or molecules) is spectrally shifted by a magnitude given by: = q v ν (2) where v is the vector velocity and q is the scattering wave-vector with magnitude equal 2π θ = sin λ 2 q (3) Figure 8: Histogram of convective velocity determined from translating template. The convective velocity is taken as the average of the four velocities calculated by displacement between image 1 and image 14-17, respectively. A criterion and where θ is the angle between the incident laser beam and the direction of the detected scattering. If a cell filled with a suitable atomic or molecular vapor is placed in front of the detector, then the fractional transmission through the cell constitutes a unique determination of the component of velocity parallel to the wave-vector. (This direction is perpendicular to the vector, which bisects the angle between the laser and the detector). If a laser beam is formed into a thin sheet, and if the filtered scattering is detected with a CCD imaging device, then instantaneous planar velocity images can be obtained. To achieve this, a pair of matched field-of-view images, one filtered and one unfiltered, are obtained and velocity field determined by pixel by pixel ratioing.

9 PDV has developed to the point where reliable velocity measurements with errors on the order of 2 m/sec are readily obtainable for a variety of flow environments. More detail can be found in recent comprehensive review articles by McKenzie 21, Elliott and Beutner 22 and Samimy and Wernet 23. As a preliminary test of the feasibility of Real Time PDV, images were obtained from an acetone seeded, high aspect ratio Mach 2 rectangular half nozzle with dimensions 1/8 x 3/4. This creates a measurement volume that is large enough to be visualized using conventional optics, but small enough that the intensity of the relatively low power laser sheet could be confined to a small region. The sheet propagated downward to the vertical jet, incident at an angle of ~ 15 degrees with respect to the streamwise direction, thus forming a quasi-streamwise view. Scattering was captured at 90 o to the jet, with a field of view of approximately 1.6 x 1.6. Figure 9 shows a picture of an experiment in progress. The laser beam is easily seen due to scattering from small (50 nm) condensed acetone particles. A cage covered in black felt held the splitter/recombiner optics and the SMD camera. The geometry of the system results in a one component PDV system that is sensitive to the 0.968i j +0.00k component of velocity, where i is the streamwise direction of the jet, j is along the major axis and k is along the minor axis of the jet. A sample raw image pair is shown in Fig. 10. (The image is slanted due to the method in which the camera acquires and reads out the image). The left-hand side of the frame contains the unfiltered image while the right-hand side contains the filtered image. This figure demonstrates the ability of the system to capture both images on a single camera, but also illustrates constraints due to the limited power available from the laser and the dynamic range of the camera. Note that the image is scaled from 0 to 140 intensity units out of a maximum of The spatial resolution of the camera also appears to be sufficient to make measurements on a variety of scales within the flow. Figure 10: Sample unfiltered (left) and filtered (right) image pair obtained with RT-PDV. Figure 11 shows a histogram of intensity ratios obtained from image pairs similar to that shown in Fig. 10. This histogram also plots a reference point taken immediately afterwards on a stationary target. Two things are clear from this figure. First, the velocity of the Mach 2 flow is clearly resulting in a bulk Doppler shift of ~0.38 dimensionless ratio units. This shift corresponds to a mean frequency shift of 732 MHz, which translates into an average streamwise velocity on the order of 400 m/sec. Second, the histogram of values obtained with the jet on is much broader than the histogram obtained with no flow. The large spread is an indication of the range of velocities being measured in the flow. Figure 9: Photograph of RT-PDV Measurement The pulse burst laser is seen flashing in the right - background. These promising, results indicate that the PDV apparatus incorporating the pulse burst laser and ultrafast camera is detecting and measuring Doppler shifted

10 scattering. Currently, however, the signal to noise ratio in the images is quite low (of order 5), dominated by large contributions of noise, thought to be the result of what is known as speckle. Speckle is a well known phenomenon, which accompanies scattering of coherent radiation, caused, fundamentally, by spatial interference that leads to random intensity minima and maxima in the image. It is well established that speckle is the dominant noise source in PDV 24 and is minimized by incorporating large camera pixels and collection lens solid angles. As discussed in more detail in Thurow, et al. 25, the relatively small pixel size (14 microns) of the framing camera employed for this study is thought to be a significant issue which needs to be addressed. We are currently exploring strategies for minimizing speckle noise, such as incorporation of a micro lens array to increase the framing camera fill factor, and testing other, larger pixel format, framing cameras. We are also planning to add an additional, ½ amplifier to the large scale structures was also developed, which employed a translating template, which convects with the structure being tracked. Convective velocity determined is much higher than the theoretical convective velocity, but closer to that of the empirical formulation of Murakami and Papamoschou. The potential feasibility of incorporating the MHz rate imaging system for extension of the Planar Doppler Velocimetry technique into the Real Time regime has been explored with a series of preliminary measurements. In particular, filtered/unfiltered image pairs were obtained from a Mach 2 rectangular nozzle. While the image quality is severely constrained by speckle noise, the experiments have demonstrated that the real time system has the potential for quantitative determination of velocity field. It is anticipated that incorporation of a camera with larger pixel format and a laser with higher individual pulse power will result in significant improvement in velocity measurement accuracy. ACKNOWLEDGEMENTS The authors would like to acknowledge the support of Positive Light, Inc, through a sub-contract from a NASA Glenn Research Center Small Business Innovative Research grant, and from the National Science Foundation and the State of Ohio Hayes Investment Fund. REFERENCES pulse burst laser, as well exploring ways to increase the energy extraction efficiency of the system. Figure 11: Histogram of transmission ratios for Mach 2.0 nozzle (right) and stationary target (left) obtained from image pairs such as that of Fig. 10. Corresponding bulk Doppler shift is 732 MHz. CONCLUSIONS A second generation MHz rate imaging system has been developed, which combines a home built Nd:YAG burst mode laser and a commercial framing CCD camera. The system has been employed to examine the flow field produced by a Mach 1.3 axisymmetric nozzle. Both streamwise and cross-stream movies were examined in order to gain a qualitative understanding of the development of large-scale structures in the jet. A novel scheme to measure the convective velocity of 1. W. R. Lempert, P-F Wu, B. Zhang, R.B. Miles, J.L. Lowrance, V. Mastracola, and W.F. Kosonocky, Pulse-Burst Laser System for High Speed Flow Diagnostics, Paper #AIAA , AIAA 34th Aerospace Sciences Meeting, Reno, NV, Jan , P Wu, W. R. Lempert, and R.B. Miles, MHz Pulse-Burst Laser System and Visualization of Shock-Wave/Boundary-Layer Interaction in a Mach 2.5 Wind Tunnel, AIAA Journal 38, 2000, pp B. Thurow, W. Lempert, and M. Samimy, Paper AIAA , 38 th AIAA Aerospace Sciences Meeting, Reno, NV. Jan 10-13, B. Thurow, J. Hileman, M. Samimy, and W. Lempert, An In-Depth Investigation of Large Scale Structures in a Mach 1.3 Axisymmetric Jet, Paper AIAA , 39 th AIAA Aerospace Sciences Meeting, Reno, NV, Jan. 8-11, W. R. Lempert, P.-F. Wu, and R. B. Miles, Filtered Rayleigh Scattering Measurements Using a Pulse-Burst Laser System, Paper #AIAA 97-

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