Tomographic Spatial Filter Velocimetry for Three-Dimensional Measurement of Fluid Velocity

Transcription

1 Tomographic Spatial Filter Velocimetr for Three-Dimensional easurement of Fluid Velocit Shigeo Hosokawa 1,*, Takaaki atsumoto 1, Akio Tomiama 1 1: Graduate School of Engineering, Kobe Universit, Kobe, Japan * correspondent author: hosokawa@mech.kobe-u.ac.jp Abstract Spatial filter velcoimetr (SFV) proposed in our previous stud are etended to threedimensional three-component (3D3) velocimetr b coupling the SFV processing with a tomographic technique. Three-dimensional particle distributions are reconstructed from images recorded b two cameras. A validation method of velocit data based on a transit time of the particle is used to reduce error velocities induced b ghost particles generated in the particle reconstruction process. The developed sstem is applied to a free jet and an impinging jet discharged from a circular pipe. The three-dimensional distributions of velocit vectors of jets are successfull measured b the developed Tomo-SFV in spite of using onl two cameras. The peak locking in velocit histograms, which is frequentl occurred in PIV measurements, dose not appear in the histograms measured b Tomo-SFV. 1. Introduction Three-dimensional three-component (3D3) velocimetr is one of challenging issues in fluid dnamics researches. Some 3D3 velocimetr [1-5] has been proposed, and applied to flows in simple geometries. Among them, tomographic particle image velocimetr (Tomo-PIV) [4] and volumetric 3-component velocimetr (V3V) [5] might be realistic methods for practical 3D3 velocimetr. These sstems consist of reconstruction of three-dimensional particle distributions and velocit evaluation from the reconstructed particle distributions. Tomo-PIV reconstructs a threedimensional particle distribution b tomograph using a set of images recorded b several cameras. Since ghost particles frequentl formed in the tomograph, particle image velocimetr based on correlation between two interrogation volumes including several particles is appropriate for velocit evaluation to reduce errors induced b presence of ghost particles. On the other hand, PIV requires multiple particles in an interrogation volume to evaluate velocit, and therefore, a high concentration of tracer particles is indispensable and it increases the number of ghost particles. Tomo-PIV, therefore, usuall requires four cameras to reduce the ghost particles. V3V reconstructs a three-dimensional particle distribution based on paralla among three apertures placed in different angles. Although this reconstruction method is effective to reduce ghost particles, it is difficult to appl it to flows with a high particle number densit. Hence, particle tracking velocimetr is frequentl used to evaluate velocit in a V3V sstem. When we appl these sstems to actual flow fields, we frequentl encounter difficult in optical arrangement because the need four or three cameras/apertures to measure velocit. Therefore, 3D3 velocimetr using onl one or two camera(s) is required to practical flow measurements. There are several reconstruction methods of three-dimensional particle destributions using one or two camera(s) such as holographic [1], defocusing [2], color gradients [3] and tomographic [4] methods. Each method has its own applicable ranges in particle size and particle number densit. If there is a velocit evaluation method, which is applicable for the wide range of particle size and particle number densit and robust against ghost particles, we can select an appropriate reconstruction method from the above mentioned techniques for a flow field. On the other hand, the authors developed spatial filter velocimetr based on time-series digital - 1 -

2 images [6]. SFV accuratel measures velocities of single tracer particles in a flow with a spatial resolution as high as LDV, and it simultaneousl measures two-components of velocit at multiple points in a plane. Since SFV measures velocities of single particles, it can be applied to the flow with dilute tracer particles in which the forming of ghost particles is suppressed in the tomographic particle reconstruction. SFV uses time-series integral brightness intensit in an interrogation area to calculate velocit, and therefore, it is robust against noises and ghost particles. SFV is applicable to flows with a high particle number densit [6] and flows with large particles [7]. Hence, SFV can be a robust velocit evaluation method for realizing 3D3 velocimetr b coupling with various kinds of three-dimensional particle reconstruction methods. In this stud, we etended SFV to 3D3 velocimetr b combining SFV with a tomographic method using onl two cameras to reconstruct three-dimensional particle distributions. We developed a prototpe sstem of tomographic spatial filter velocimetr (Tomo-SFV), and applied it to a jet discharged from a circular pipe to eamine its feasibilit and potential as robust 3D3 velocimetr. 2. Principle of Tomographic Spatial Filter Velocimetr 2.1 Principle of two-dimensional spatial filter velocimetr Laser Doppler velocimetr (LDV) forms fringes in its measurement volume b interference of two beams irradiated from an LDV probe, and detects flashing light scattered from a tracer particle locating in the measurement volume. Then, it evaluates the frequenc of the flashing light to measure the velocit of a particle following the flow. Spatial filter velocimetr (SFV) [6] detects flashing signal generated b appling a spatial filter to time-series particle images recorded b a video camera, and evaluates the frequenc of the flashing signal to measure the particle velocit. Hence, SFV can be regarded as software LDV. Figure 1 shows the principle of SFV. Tracer particles added to a flow are illuminated b a light source such as a laser light sheet, and time-series particle images are recorded b a video camera (image acquisition step). Intensit distribution I (,, in an interrogation area (measurement region) in each particle image is multiplied b a spatial filter function F SF (,,, and the integral intensit I SF ( of the spatiall-filtered I(,, is calculated at each recording time (spatial filtering step): SF I ( (,, dd (1) = I(,, FSF where t is the time and and are the coordinates in the interrogation area. Note that the noise level in I SF ( is lower than that in I(,, due to the spatial integration of random noise in I(,,. When a particle is moving in direction with the velocit v, the integral intensit I SF (, which is calculated b using a periodic spatial filter function F SF () with the wavelength L in direction, periodicall fluctuates with the frequenc f = v /L as shown in Fig. 1. Hence, the particle velocit v can be measured b evaluating the frequenc f of the integral intensit I SF ( (frequenc/velocit evaluation step). The other velocit component v, which is perpendicular to v, can be measured b using a periodic spatial filter function F SF () with the wavelength L in direction. Thus, we can evaluate magnitudes of two velocit components of each tracer particle in an arbitrar interrogation area in the imaging plane from the time-series particle images. However, the direction of the velocit cannot be evaluated b this method, and velocit measurement is difficult for stationar particles. These problems can be solved b moving a spatial filter function F SF ( i, (frequenc - 2 -

3 shifting), where i is the coordinate in the moving direction. This method is similar to the frequenc shifting in the LDV measurement. Details on the two-dimensional SFV can be found in Hosokawa and Tomiama [6]. ain features of SFV are (1) velocit of a single tracer particle is measurable, so that accurac, spatial and temporal resolutions can be as high as LDV, (2) simultaneous measurement of two velocit components is possible at arbitrar multiple points in recorded particle images, (3) it is applicable to flows with a low concentration of tracer particles, (4) processing conditions such as an interrogation area, a pattern of spatial filter, a shift frequenc and a frequenc evaluation method can be easil changed after recording particle images, and (5) SFV can measure translational velocit of a pattern such as bubble and wave velocities like PIV [7]. Light sheet amera -series particle images Flow + particles Image acquisition step I L X I ( t = F dd SF ) easurement region SF v Spatial filter I (,, t ) F SF (,, t ) Spatial filtering of particle image alculation of integral intensit of filtered image Spatial Filtering step Integral intensit I SF ( t ) f = 1/ t = V X /L X t Frequenc f (Velocit V ) Frequenc analsis V X = L X / t = L X f Frequenc/velocit evaluation step Fig. 1 Principle of spatial filter velocimetr 2.2 Etension of SFV to 3D3 measurement Principle of Tomo-SFV is illustrated in Fig. 2. ajor modifications for realizing Tomo-SFV are etension of the image acquisition step to 3D, addition of a step of reconstructing a threedimensional particle distribution and etension of the spatial filtering step to 3D

4 For image acquisition and 3D reconstruction steps, we can make use of techniques used in 3D3 PIVs and 3D PTVs [1 4] such as a tomographic technique and a V3V technique. Since a few cameras are required for application to practical flows, we reconstructed particle distributions using onl two cameras based on the tomographic technique [4]. Tracer particles added to a flow are illuminated b a light source, and time-series particle images, I( 1, 1, and I( 2, z 2, (the subscripts indicate the camera number), are recorded b two highspeed cameras from perpendicular directions along z and aes (image acquisition step). A preconditioning process, that is, background subtraction [8] and normalization of brightness using the maimum and the minimum intensities, has been applied to I( 1, 1, and I( 2, z 2,. Threedimensional particle distributions I(,, z, are calculated from I( 1, 1, and I( 2, z 2, b the tomographic method (particle reconstruction step): I,, z, = I(,, I(, z, ) (2) ( t where i, i and z i are the coordinates on image planes recorded b camera i. The relationship between (,, z) and ( 1, 2, 1, z 2 ) is given b 1 2 = 1 z z z2 1z 2z 1z z2z A A + A z A z2 (3) where is the magnification of the image and A the offset of the ais on the image plane. The values of and A are calculated from images of a calibration target located in the measurement region, which is recorded before velocit measurement, b using a least square fit. To obtain a velocit component in i direction, the 3D particle distributions in an interrogation volume ( + /2, + /2, z + z/2) are spatiall filtered b a filter function F SF ( i,, and the filtered intensit distribution, I(,, z, F SF ( i,, is integrated over the interrogation volume (spatial filtering step): z + z I SF, i(,, z, = I(,, z, FSF ( i, dddz (4) z z where I SF,i is the integral intensit of spatiall filtered I in i direction. When a particle is moving in i direction with the velocit v i, the integral intensit I SF,i, which is calculated b using a periodic spatial filter function F SF ( i, with the wavelength L i in i direction, periodicall oscillates with the frequenc f i = v i /L i as shown in Fig. 2. Hence, the particle velocit v i can be measured b evaluating the frequenc f i of the integral intensit I SF,i ( (frequenc/velocit evaluation step). Each velocit component can be measured b using a spatial filter function F SF ( i ) with the wavelength L i in each direction. Thus, we can evaluate three velocit components of each tracer particle in an arbitrar interrogation volume in the measurement region from the time-series reconstructed particle distributions. The direction of the velocit can be measured b using a moving filter function like the frequenc shifting in LDV

5 Reconstructed particle motion in an interrogation volume Spatial filtering in direction 2D particle image 2D particle image in -z plane in - plane I(, z, I(,, amera 2 easurement region Interrogation region 3D reconstructed particle z I(,, z, amera 1 Image acquisition & reconstruction step L Integral intensit in direction: I SF, ( Spatial filtering in direction L Integral intensit in direction: I SF, ( Spatial filtering in z direction t = 1/f t = 1/f Spatial filtering step L z Integral intensit in z direction: I SF,z ( t z = 1/f z Wavelet analsis of I SF in each direction f i f D,i v i = L i (f D,i f S,i ) High P W * Low Frequenc/velocit evaluation step Fig. 2 Principle of tomographic spatial filter velocimetr

6 3. Prototpe sstem and its application to a jet flow 3.1 Eperimental apparatus and measurement sstem The developed Tomo-SFV was applied to a water jet discharged from a circular pipe. The inner and outer diameters of the nozzle were 5 and 7 mm, respectivel. The nozzle eit was placed at the center of the test cell, the size of which was 5, 5 and 25 mm in width, depth and height, respectivel. The water was filled in the upper tank and flowed into the test cell through the nozzle b the hdro-static pressure difference between the upper tank and the test section. Flow rate was measured from change in height of water surface in the test cell. The area-averaged velocit in the pipe was.1 m/s. The water temperature was fied at 25 degree elsius through the eperiments. Eperiments were carried out for a free jet and an impinging jet on a flat plate. In the free jet condition, the nozzle tip was located at 2 mm from the bottom of the test cell, and the velocit distribution was measured in the rectangular region ( mm) just below the nozzle eit as shown in Fig. 4 (a). The flat plate was placed at 5.5 mm below the nozzle eit in the impinging jet condition, and the velocit distribution was measured in the rectangular region (9 9 5 mm) shown in Fig. 4 (b). We developed a prototpe Tomo-SFV sstem which consists of a DPSS laser (Giga Laser, λ = 532 nm, 1 W) with optics, two snchronized high-speed cameras (DANTE, NonoSense kiii) and processing software based on the above-mentioned methodolog. A cosine function was used for the spatial filter F SF, and a wavelet analsis was used to evaluate the frequenc f i. The details of the frequenc evaluation method can be found in Hosokawa and Tomiama [6]. A calibration target with circular dots (diameter:.1 mm, pitch: 2 mm) was recorded b the cameras to calculate the values of and A in Eq. (3) before each eperiment. Polstrene particles of 2 µm diameter were added to the water. The tracer particles were illuminated from the bottom of the test cell b a beam emitted from the laser, which was epanded to 2 25 mm in diameter b lenses. The two highspeed cameras (frame rate: 8 fps, resolution: 55 µm/piel) recorded the particle images from the perpendicular directions. The velocities were calculated from 1, frames at each voel ( piel), and the sample number of velocit at each voel was about 1,. The wavelength of the filter was set at 3 piels. Upper tank Flow Valve 2 15 mm Valve1 High speed camera 2 High speed camera 1 5 mm (a) free jet 5 mm z lens2 lens1 Laser Fig. 3 Eperimental setup 5 mm Flow 5.5 mm 9 mm 9 mm Acrlic plate (b) impinging jet Fig. 4 easurement regions - 6 -

7 3.2 Results and discussion Eamples of I SF ( and its spectrogram calculated b the wavelet transform are shown in Fig. 5. The red curves in the figures represent integral intensit I(,, z, of non-filtered intensit I (,, z, : z + z I (,, z, = I(,, z, dddz (5) z z I SF ( of a reconstructed correct particle passing through an interrogation volume shows periodic oscillation as shown in Fig. 5 (a), and its spectrogram takes a sharp peak around 2 Hz which corresponds to the fluid velocit. This signal shows the successful reconstruction of the particles and successful velocit measurement in the present sstem. Ghost particles were sometimes formed in the reconstruction step, though the number of ghost particles was about 1% of the number of correct particles in the present eperimental range. ost of the ghost particles appeared in a short duration as shown in Fig. 5 (b), and therefore, it can be distinguished from the correct particles b checking the time duration of high I(. In this stud, the signal, which was shorter than 2 % of the transit time (the interrogation size divided b the veloci, was rejected from velocit data. Figure 5 (c) shows I SF ( and spectrogram when a ghost particle appears in an interrogation area in which a correct particle is passing through. The peak in the spectrogram, which corresponds to the signal from the correct particle, is not affected b the presence of the ghost particle. Although there is a weak peak in a high frequenc region in the duration of appearance of the ghost particle, this peak is lower than the peak of the signal from the correct particle. Thus, the overlapping of signals from correct and ghost particles does not affect measured velocit. It should be noted that this overlapping rarel occurs. Ghost particle I SF ( ISF( I SF ( -1.1 t [s] -1.5 t [s] t [s] f [Hz] f [Hz] f [Hz] t [s] t [s] t [s] (a) without ghost particle (b) a ghost particle (c) with ghost particle t [s] Fig. 5 Eamples of integral intensit I SF and its spectrogram (upper: I SF, lower: spectrogram) - 7 -

8 A measured three-dimensional mean velocit field of the free jet discharged from the nozzle is shown in Fig. 6, and two-dimensional velocit distribution at each vertical plane is shown in Fig. 7. The aial velocit takes a high value at the center ( = 4 mm, z = 4 mm) of the nozzle eit and it decreases as increases and as being awa from the center in or z direction. The velocit field is more or less aismmetric. The three-dimensional distribution of velocit is successfull measured b the developed Tomo-SFV in spite of using onl two cameras. Figure 8 shows the histograms of aial velocit on the center line of the jet. It is well known that the peak locking is apt to occur in velocit histograms measured b PIV. To the contrar, the peak locking does not appear in the histograms measured b SFV. This implies that SFV is a more accurate measurement method than PIV. The center velocit of the histogram decreases and the width of the histogram become wider with increasing /D (D: inner diameter of the nozzle). The measured velocit distributions were compared with the velocit distribution of a laminar jet discharged from a circular hole given b [9] where 3 K 1 u = (6) 8π ν ( 1+ ξ 2 4) 2 1 ξ = 4 K = 2 3 π π u K ν 2 r rdr (7) U [m/s] (a) velocit vectors (b) streamline Fig. 6 easured velocit distribution and streamline of a jet discharged from a circular pipe

9 [mm] [mm] [mm] [mm] [mm] [mm] z [mm] z [mm] - plane (z = 3 mm) - plane (z = 4 mm) -z plane ( = 3 mm) -z plane ( = 4 mm) Fig. 7 Velocit distributions. PDF PDF U [m/s] U [m/s] (a) /D =.2 (b) /D = 1. PDF PDF U [m/s] U [m/s] (c) /D = 1.5 (d) /D = 2. Fig. 8 Histograms of aial velocit - 9 -

10 Figure 9 shows comparisons between measured aial mean velocit and Eq. (6). The horizontal ais is the distance r from the center of the jet, and θ the angle from the ais in the horizontal plane. The measured aial mean velocit agrees well with Eq. (6) at each /D. Note that the variation in the measured velocit distribution might be mainl due to weak asmmetr and fluctuation of the jet. These results clearl demonstrate a substantial potential of Tomo-SFV in 3D3 measurement and applicabilit of SFV to the evaluation of three-dimensional particle velocit..1 /D =.5 U[m/s] r[mm].1 /D = 1. /D = U[mm] r[mm] U[mm].5 θ<36 36 θ<72 72 θ<18 18 θ< θ< r[mm] Fig. 9 Distributions of aial mean velocit of free jet A three-dimensional velocit distribution in the impinging jet measured b Tomo-SFV is shown in Fig. 1. The radial distribution of the radial velocit component V r near the wall surface is plotted in Fig. 11. The velocit distribution clearl represents that the jet discharged from the nozzle impinges on the wall and spreads in the radial direction. The measured velocit distribution is aismmetric not onl in the jet region but also in the radiall spreading region as shown in Fig. 11. This result clearl shows that Tomo-SFV successfull measures a three-dimensional distribution of three components of velocit. Thus, we confirmed that Tomo-SFV has a potential of 3D3 velocimetr in practical flow fields with simpler sstem configuration than Tomo-PIV and V3V sstems. 4. onclusion Spatial filter velcoimetr (SFV) was etended to three-dimensional three-component velocimetr b coupling the SFV processing with a tomographic technique. Three-dimensional particle distributions were reconstructed from images recorded b two cameras. A validation method of velocit data based on a particle transit time was adopted to reduce error velocities caused b ghost particles formed in the particle reconstruction process. The developed sstem was applied to a free - 1 -

11 jet and an impinging jet discharged from a circular pipe. The three-dimensional distribution of velocit vectors of the jets were successfull measured b the developed Tomo-SFV in spite of using onl two cameras. The peak locking, which is apt to occur in velocit histograms measured b PIV, did not appear in the histograms measured b Tomo-SFV. The results demonstrated a substantial potential of Tomo-SFV in 3D3 velocimetr and applicabilit of SFV to the evaluation of three-dimensional particle velocit. Flow U [m/s] - plane z -z plane z Wall Fig. 1 easured 3D velocit distribution of the impinging jet V r [m/s].2.1 θ<36 36 θ<72 72 θ<18 18 θ< θ< r[mm] Fig. 11 Radial velocit distribution near the wall ( = 4.5 mm)

12 Acknowledgement The authors gratefull acknowledge r. Kosuke Kitahata (Kobe Universi for his assistance to eperiments. This work has been partl supported b the Japan Societ for the Promotion of Science (grants-in-aid for scientific research () No ). References [1] Zhang, J., Tao, B. and Katz, J., 1997, Turbulent Flow easurement in a Square Duct with Hbrid Holographic PIV, Eperiments in Fluids, Vol. 23, pp [2] Pereira,., Gharib,., Dabiri, D. and odarress, D., 2, Defocusing Digital Particle Image Velocimetr: A 3-omponent 3-Dimensional DPIV easurement Technique. Application to Bubbl Flows, Eperiments in Fluids, Vol. 29 Suppl. 1, S78- S84. [3] Hosokawa, S. and Tomiama, A., 27, A Three-Dimensional PIV using Intensit Gradients of a Two-olor Laser Beam, ultiphase Science and Technolog, Vol. 19, No. 3, pp [4] Elsinga, G.E., Scarano, F. and Wieneke, B., 26, Tomographic Particle Image Velocimetr, Eperiments in Fluids, vol. 41, pp [5] Troolin, D. R. and Longmire, E. K., 21, Volumetric Velocit easurements of Vorte Rings from Inclined Eits, Eperiments in Fluids, Vol. 48, pp [6] Hosokawa, S. and Tomiama, A., 212, Spatial Filter Velocimetr based on -Series Particle Images, Eperiments in Fluids, Vol. 52, No. 6. [7] atsumoto, T., Hosokawa, S. and Tomiama, A., 212, easurements of Bubble Velocit using Spatial Filter Velocimetr, Proc. 16th International Smposium on Applications of Laser Techniques to Fluid echanics, Lisbon, Portugal. [8] Honkanen,. and Nobach, H., 25, Background Etraction from double-frame PIV images, Eperiments in Fluids, Vol. 38, pp [9] Schlichting, H., 196, Boundar Laer Theor, cgraw Hill