Sep 27, 1996 - M. Raffel, C. Willert, M. Rosengarten, J. Kompenhans. Institut fu¨r Stro¨mungsmechanik, Deutsche Forschungsanstalt fu¨r Luft- und Raumfahrt ...
Experiments in Fluids 23 (1997) 331—334 ( Springer-Verlag 1997
Hybrid stereoscopic particle image velocimetry M. Gaydon, M. Raffel, C. Willert, M. Rosengarten, J. Kompenhans
331 Abstract The technical aspects of a photographic stereo camera for three-dimensional particle image velocimetry are described herein. The hybrid concept of the camera combines advantages of the angular displacement and the translation method. The camera uses two CCD sensors in order to adjust the lens distances and angles to meet the Scheimpflug criterion and two coupled rotating mirrors for image shifting. An application to a jet flow with an exit velocity of 33 m/s demonstrates the succesfull optimization of the recording process.
1 Introduction Stereoscopic particle image velocimetry (PIV) allows for the measurement of three-dimensional velocity vectors in a two dimensional plane. Stereoscopic PIV also adds the advantage of eliminating perspective error which exists in conventional PIV in the presence of an out-of-plane component of velocity. For a pair of lenses in a stereoscopic arrangement with identical focal length f, increasing the viewing angle will increase the accuracy of the measurement of the out-of-plane component of velocity. All three components of velocity can be measured with similar accuracy at large viewing angles. However, this requires taking added precautions against defocusing. Gauthier and Riethmuller (1988) present two different methods to produce oblique viewing angles H. The translation method uses image, object, and lens planes that are all parallel to each other. The separation of the two lenses provides
Received: 27 September 1996 /Accepted: 6 March 1997 M. Gaydon Dep. of Mech. Engineering Lehigh University 354, Packard Laboratory 19 Memorial Drive West Bethlehem, PA 18015-3085, Pennsylvania, USA M. Raffel, C. Willert, M. Rosengarten, J. Kompenhans Institut fu¨ r Stro¨ mungsmechanik, Deutsche Forschungsanstalt fu¨ r Luft- und Raumfahrt (DLR), Bunsenstraße 10, D-37073 Go¨ ttingen, Germany Correspondence to: M. Raffel The authors would like to thank IAESTE that supported the visit of Matthew Gaydon at DLR in Go¨ ttingen.
the oblique viewing angle. In addition, the back planes are offset with respect to the lenses so that both cameras image a common area in the object plane. The angular displacement method uses tilted lenses to provide the oblique viewing angle of a common field of view. The image plane must also be tilted to satisfy the Scheimpflug criterion so that the image is well focused over the whole observation area (Hinsch 1993). Our experimental camera also implemented image shifting via rotating mirrors located between the lenses and the film planes to eliminate the directional ambiguity of the velocity measurements. This image shifting technique allows stereoscopic PIV to be applied to flows with large out of plane velocities and/or areas of reverse flow.
2 Comparison of the techniques The translation method is more commonly used than the angular displacement method because it is simple to apply. Image distortions are negligible and conventional evaluation algorithms can be used to interrogate the images. The main disadvantage of this method is that the viewing angle is limited by the quality of the lenses. At large viewing angles with respect to the optical axis, the particle image size increases due to astigmatic aberrations (Hecht and Zajac 1974). This causes a more or less significant drop of the modulation transfer function (MTF). For very small particles, where the geometric image diameter can be neglected, the following formula can be used to estimate the resulting particle image diameter d as i a function of the modulation transfer for different viewing angles (Raffel and Kompenhans 1996):
S
d (H)+ 0.64 i
A B
M 2 [ln(MTF(r@, H)) [ , 2r@ r@2
where MTF(r@, H) is the modulation transfer for a certain viewing angle at the spatial frequency r@, and M is the average magnification (M\image size/object size). If MTFs are given for different r@, a spatial frequency which is suitable for image reconstruction should be taken into account (r@ · d +1). i The increased particle image diameters that are encountered at large viewing angles reduce the obtainable spatial resolution, since larger particle image diameters require larger pulse separation times and larger interrogation windows for similar measurement accuracy. This can be quantified by the translation efficiencies plotted in Fig. 1, which are the ratios of the minimum image diameters for H\0° and the image diameters estimated for a certain viewing angle H[0° for different
332
f-numbers ( fd). For example (for fd\22): N22\ d (0°)/d (H). The data represented in Fig. 1 were computed i i for 35 mm film format using the MTF of the lenses which were used in the experiments (see Sect. 3). These lenses are optimized for large film formats and provide a much higher image contrast at large viewing angles than lenses for SLR cameras normally do. An ideal lens system would have a modulation transfer function with a relatively small slope (small variation in particle image size) for the range of angles that is to be used. Typically the modulation transfer function becomes more favorable as the aperture is reduced; however, the imaging of the particles will then require more laser power. In the angular displacement method of stereoscopic PIV the only factors that limit the viewing angle are the mechanical limits of the camera. However, special software must be applied when correlating the images to account for the variation in magnification. Moreover, the variation of magnification and a square or rectangular sensor or film format leads to an observation area which can best be described as trapezium. Therefore, an area of the image plane is lost or wasted because it can not be matched to a corresponding area on the image from the other camera. When using photographic techniques large format films can be used to compensate for the lost area. The loss of area is more crucial when a digital PIV system is used as additional image area is very expensive. This fact can be expressed by calculating the ratio of minimum magnification M @(H ) and M, the average 5*-5 magnification (image size/object size):
The hybrid stereoscopic PIV method uses a combination of the two methods that have been mentioned thus far. The amount of translation and angular displacement are determined by maximization of an efficiency function. A plot of the hybrid efficiency, which is the product of the translation efficiency ( fd\22) and the tilt efficiency, versus the tilt angle, is represented by the thick dashed and dotted line in Fig. 1. It was computed for an angle of 45° between the viewing directions of both cameras Nh\N22 (22.5°[H )Nt(H ) 5*-5 5*-5 (viewing angle of each camera H\22.5°). The comparison of the different curves shows that the angular displacement method or a combination of a large angular displacement and a relatively small lateral translation offer best results.
3 Camera
where Z is the distance between the lenses and the film plane, i f is the focal length, x@ the sensor width (35 mm) and, a the angle between film plane and lens plane determined by the Scheimplug criterion: M\tan a/tan H.
A stereoscopic camera with two rotating mirrors employing the translation method was proposed by Raffel et al. (1992). A camera following these ideas was constructed in order to apply it for the investigation of the wake of cylinders (Seyb 1996). During the test of the camera severe problems of the optical quality of the recordings were encountered. This motivated the hybrid stereoscopic method as described herein. Two identical Rodenstock Grandagon 90 mm lenses were used for the stereoscopic camera system. The lenses were mounted such that it was possible to move them in a combination of translational and angular motions. Bronica 35 mm film boxes were mounted on the film plane for photographic recording (Fig. 2). The film planes were initially located at 90° with respect to the lens planes. A rotating mirror — at 45° during exposure — was used to reflect the image to the film. The film plane was designed so that it could be tilted with respect to the lens to satisfy the Scheimpflug criterion. In order to focus the image on the film, two standard video chips — with high spatial resolution due to a small sensor size — temporarily replaced the film. The lenses and film planes were adjusted until the particle image size was minimized over the extent of the film area. This procedure provided a means to
Fig. 1. Resolution efficiency for translation method at different fnumbers, angular displacement method, and a combination of both methods
Fig. 2. Diagram of the hybrid stereoscopic camera using photographic film (35 mm) and coupled rotating mirrors for image shifting
Nt\M @(H )/M (see Fig. 1). 5*-5 Z [f[sin(a)x@/2 M @(H )\ i 5*-5 f
experimentally adjust the lenses and film planes to satisfy the Scheimpflug criterion thereby guaranteeing that the images on the film are in the best possible focus before any photographs were taken. The rotating mirror provides for the possibility of relatively high velocity image shifting. In the past, image shifting was applied to stereoscopic PIV by mechanically translating the film plane between particle exposures (Prasad and Adrian 1993). The rotating mirror system allows for much larger shift velocities, thereby allowing stereoscopic PIV to be applied to a broader range of flows. The axes of the mirror are coupled by a twist free shaft clutch; therefore, the two mirrors were perfectly synchronized.
333
4 Evaluation The recordings were scanned with a commercially available slide scanner at 2700 dpi. The image shift due to the rotating mirror has been removed in the way described by Raffel and Kompenhans (1996). In order to determine the transformation matrices a cross-correlation between the two distorted, doubleexposure recordings has been computed. The transformation used to remap the distorted recordings can best be described as the digital counterpart of the back-projection scheme as presented by Hinsch (1995). Then the displacement vectors were computed by local cross-correlation of two interrogation windows of the same recording, which were displaced by the mean image shift, as described by Willert (1996). The pair of two-dimensional data sets are then merged into one threedimensional data set using the distance between the lenses and the distance to the light sheet (see e.g. Arroyo and Greated 1991). Finally, the velocity vectors were obtained from the time delay between the two light pulses and a remapped calibration recording from which the magnification factor was taken. The overall uncertainty of the out-of-plane velocity component measurement was estimated to be below 2% of the maximum flow velocity.
5 Experiment A simple experiment was used to test the experimental camera. A jet of air and its surrounding was seeded with 1 lm oil droplets and a cross-section was illuminated by laser light pulses and photographed from two different viewing angles. Each camera viewed the light sheet from an angle of H+18° with respect to the axis of symmetry of the camera system (H \13.7°, 4.5° due to translation). The camera system was 5*-5 located at distance of 330 mm from the light sheet (measured perpendicular from the front of the lenses) and had an average magnification of M\0.34. The resulting common field of view was approximately 100 mm]70 mm. At the viewing location the laser sheet was approximately 0.5 mm thick. The area of interest was illuminated from a direction perpendicular to the plane containing the optical axes of the cameras. This direction did not provide the best light scattering intensity, but the light scattering intensity was similar for both cameras. The use of a relatively high powered pulse laser (300 mJ per pulse) and the relatively small observation field allowed this procedure. The images were recorded on high-sensitivity black and white film (Kodak TMAX 3200) and were interrogated using the cross-
Fig. 3. Three-dimensional representation of the instantaneous velocity field of a 33 m/s jet flow recorded 50 mm from the nozzle, which had an orifice diameter of 25 mm
correlation technique. An air jet flow was viewed at a distance of two nozzle diameters (d \25 mm) from the exit of the n nozzle. The light sheet was vertically oriented with the axis of the nozzle nearly orthogonal to it (\3° ascending). The jet velocity as determined by pressure and temperature measurements was 33 m/s at the exit of the nozzle. This value is in good correspondence to the out-of-plane velocities found in the inner plateau which is representing the potential core of the jet. A two-dimensional velocity vector plot with iso-velocity lines of the out-of-plane component (labeled with values in m/s) can be seen in Fig. 3.
6 Conclusions Design aspects of a stereo camera for three-dimensional particle image velocimetry on planar domains have been described. The camera has been optimized for the measurement of high speed air flows. The comparison of different methods shows the superior features of the angular displacement method with respect to the translation method as already stated by other authors (e.g. Hinsch 1995, Prasad and Jensen 1995). An application to a jet flow with an exit velocity of 33 m/s demonstrates the succesfull optimization of the hybrid method. The digital back-projection allows an easy adaptation to software already available for conventional PIV evaluation. In near future the use of modern CCD camera architectures should allow a considerably reduced experimental effort when setting up and aligning stereo cameras. Their recordings can then be evaluated by the same algorithms.
References
334
Arroyo MP; Greated CA (1991) Stereoscopic particle image velocimetry. Measurement, Science, and Technology 2: 1181—1186 Gauthier V; Riethmuller ML (1988) Application of PIDV to complex flows: measurements of the third component. VKI-LS 1988-06 Particle Image Displacement Velocimetry (Von Karman Institute for Fluid Mechanics, Rhode-Saint-Genese, Belgium) Hecht E; Zajac A (1974) Optics. Addison-Wesley, Reading MA, pp 180—182 Hinsch KD (1993) Particle image velocimetry. Speckle Metrology (Ed. Sirohi, R.; S.) Marcel Decker, New York, Basel, Hong Kong Hinsch KD (1995) Three-dimensional particle image velocimetry. Meas Sci Technol 6: 742—753 Prasad AK; Adrian RJ (1993) Stereoscopic particle image velocimetry applied to liquid flows. Exp Fluids 15: 49—60
Prasad AK; Jensen K (1995) Scheimpflug stereocamera for particle image velocimetry in liquid flows. Appl Opt 34: 7092—7099 Raffel M; Kompenhans J; Ho¨ fer H (1992) Vorrichtung zur dreidimensionalen Bestimmung von Stro¨ mungen. Patent pending 06.11.1992, and Apparatus for the three-dimensional determination of flows. US Patent No. 5,400,144, Aug. 8. 1995 Raffel M; Kompenhans J (1996) Theoretical and experimental aspects of piv recording utilizing photographic film and mechanical image shifting. VKI-LS 1996-03 Particle Image Velocimetry (Von Karman Institute for Fluid Mechanics, Rhode-Saint-Genese, Belgium) Seyb H (1996) Untersuchung der Strukturen in der Nachlauftransition eines quer angestro¨ mten Zylinders mit Hilfe von Feldmeßverfahren. Dissertation Universita¨ t Go¨ ttingen Willert CE (1996) The fully digital evaluation of photographic piv recordings. Appl Sci Research 56: 79—102
.
