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Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography
Optics & Laser Technology 31 (1999) 53±65
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Quantitative visualization of high-speed 3D turbulent ¯ow structures using holographic interferometric tomography B.H. Timmerman a, 1, D.W. Watt b, 2, P.J. Bryanston-Cross c,* a
Department of Aerospace Engineering, Delft University of Technology, Laboratory for High Speed Aerodynamics, P.O. Box 5058, 2600 GB Delft, Netherlands b Department of Mechanical Engineering, The University of New Hampshire, Kingsbury Hall, Durham, NH 03824, USA c Optical Engineering Laboratory, Department of Engineering, University of Warwick, Coventry CV4 7AL, UK
Abstract Using holographic interferometry the three-dimensional structure of unsteady and large-scale motions within subsonic and transonic turbulent jet ¯ows has been studied. The instantaneous 3D ¯ow structure is obtained by tomographic reconstruction techniques from quantitative phase maps recorded using a rapid-switching, double reference beam, double pulse laser system. The reconstruction of the jets studied here reveal a three-dimensional nature of the ¯ow. In particular an increasing complexity can be seen in the turbulence as the ¯ow progresses from the jet nozzle. Furthermore, a coherent three-dimensional, possibly rotating, structure can be seen to exist within these jets. The type of ¯ow features illustrated here are not just of fundamental importance for understanding the behavior of free jet ¯ows, but are also common to a number of industrial applications, ranging from the combustion ¯ow within an IC engine to the transonic ¯ow through the stages of a gas turbine. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Tomography; Interferometry; Turbulent structures
1. Introduction The number of quantitative ¯ow investigation tools available in engineering has increased rapidly over recent years thanks to fast developing technologies. This has made it possible for some techniques, such as laser Doppler and Laser-2-focus, to mature from being laboratory research instruments to becoming commonplace industrial tools. However, although these techniques provide quantitative measurements, they remain essentially point measurement devices. Thus, they require signi®cant data acquisition times to map a whole ¯ow ®eld which makes them a poor tool for * Corresponding author. Tel.: +44-1203-523-131; fax: +44-1203418-922. 1 Tel.: +31-15-278-5349; fax: +31-15-278-7077. E-mail address: B. [email protected] (B.H. Timmerman) 2 Tel.: +1-603-862-25555; fax: +1-603-862-2486.
globally mapping such features as the turbulent structure within an unsteady transonic ¯ow. To overcome this limitation whole ®eld instruments like PIV [1] and global Doppler velocimetry [2] are emerging as industrial tools. However, these methods are still both highly dependent on the need to `seed' the ¯ow ®eld with particles. In several cases the nature of the seeding may be a prohibitive limitation. This is the case for instance when there are ¯uctuations in a ¯ow which are in the high kHz range, like those found in structures within turbulent transonic jet ¯ows. For the seeding to follow the ¯ow accurately in these cases, the particles need to be smaller than 0.2 mm in size. Bigger particles may then give erroneous information. This is also important when `following' a travelling shock wave, which again demands a very fast inertial response of the seeding particulate. Also, the nature of the ¯ow studied may inhibit the use of seeding. Examples of this are the highly volatile regions such as found in combustion ¯ows, where the life-time of a
0030-3992/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 9 ) 0 0 0 3 6 - 5
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small particle is `short' or ¯ows where there is a concentration of other larger particle debris in the ¯ow. Furthermore, in some cases the introduction of a particulate can create a signi®cant perturbation to the ¯ow ®eld. In these cases there is a need for a technique that requires no seeding particles to make a measurement of the ¯ow. Digital holographic interferometry (DHI) [3] uses properties inherent to the ¯ow, like temperature or density dierences, thus enabling a nonintrusive, instantaneous study of such compressible ¯ows. The drawback to interferometric techniques is that they provide line-of-sight measurements. Because the majority of compressible ¯ow features of interest are three-dimensional, a method is required to extract local information from the projected data. The holographic interferometric tomography (HIT) method described here uses a number of multiple views which are formed simultaneously and stored holographically to tomographically reconstruct this local 3D data. Thus, both the large and smaller scale turbulence structures of a transonic jet ¯ow may be visualized in three dimensions. Over recent years many attempts have been made to extract three-dimensional data using interferometric tomography. As the method is numerically intensive and generally uses underde®ned data it has tended to yield low resolution and uncertain representations of the ¯ow ®eld. By combining the quantitative accuracy, as is obtained from automatically phase unwrapped interferometric images and a higher resolution stable tomographic solution this problem has been overcome. This paper shows how a high resolution tomographic solution can provide a detailed description of the complex multi scale structure and its spatial development found within a transonic jet ¯ow. Several speci®c features, like the three-dimensional coherent structure of the jet have been visualized using the HIT process which would have not been possible using other techniques. Also the structure of the turbulence and its spatial development can be seen. Further by making dierential interferograms at dierent time intervals the timescale, shape, strength and velocity of the structures can be determined. Such observations may be very relevant in studying the mixing rate and eciency in the combustion process and stage ef®ciency in turbomachinery. Furthermore to further develop numerical three-dimensional unsteady transonic ¯ow simulation models, calculations obtained by these models have to be evaluated against the type of data shown in this paper. Thus, a better understanding of the underlying processes may be obtained, which in turn creates new and more ecient power generation systems. The interferometric set-up presented performs an in-
stantaneous and simultaneous recording of multiple projections at dierent angles. From these projections the instantaneous 3D density (¯uctuation) ®eld can be obtained using tomographic reconstruction techniques. The performance of the HIT system was assessed for several types of compressible free jet ¯ows.
2. Holographic interferometric tomography 2.1. Holographic interferometry Holographic interferometry is a technique that may be used to study the refractive index distribution in transparent media. By sending interrogation beams through the region of interest optical ray path integrals, representing the refractive index ®eld are obtained in the form of phase maps. For compressible ¯ow ®elds the refractive index is related to density by the Gladstone±Dale relation [4]. Ignoring refraction, this relation can be used to relate the gas density directly to a phase change of the light ray passing through it [5]: 2pK r x, y, z ÿ r0 dz 1 Df x, y l where l is the wavelength of the probe beam, r is the density distribution, r0 is the (constant) background/ reference density ®eld, Df is the phase dierence between a beam passing through the ¯ow ®eld and one which only passes the background ®eld and K is the Gladstone±Dale constant, which is about 0.225 103 m3/kg for air at a probing wavelength of 694 nm. The phase data used in this paper is obtained from a two-reference beam, plane wave, tomographic holographic interferometer which uses a ruby pulse laser to holographically store the phase distribution associated with the ¯ow ®eld [3,6]. Unsteady ¯ows may be studied by operating the laser in the `Q' switched mode, giving 30 ns duration laser pulses which eectively freeze the motion of the ¯uid even at high ¯uctuation rates. The system allows separate recording of two ¯ow conditions at an adjustable microseconds timeinterval using the laser in double-pulsed mode, replaying of the holograms in exactly the same set-up using a HeNe laser and spatial ®ltering to remove undesirable cross-reconstructions. Thus, errors due to optics or badly repositioning of the hologram are avoided while speckle-free, plane-wave interferograms are generated which are stored using a CCD camera (512 512 pixels) [7]. Both the double exposure and double-pulsed interferograms are processed using phase-stepping techniques [6]. The 2p phase jumps in the wrapped phasemaps are then removed using phase-unwrapping [8] to
B.H. Timmerman et al. / Optics & Laser Technology 31 (1999) 53±65
Fig. 1. Projection-imaging geometry.
obtain the continuous phase data at an accuracy of approximately 1/10th of a fringe [6]. The double exposure method allows a quantitative measurement of both the instantaneous compressible ¯ow and the rapidly changing unsteady density ¯uctuations [7]. 2.2. Tomography The purpose of optical tomography is to obtain the 3D distribution of a scalar quantity from a set of projections like those obtained in compressible ¯ow interferometry. There, spatially resolved 3D density information can be obtained by tomographically reconstructing the data from a series of ray path projections along dierent viewing angles through a ¯ow ®eld. HIT, for example, can be used to measure turbulent ¯ows, which are both unsteady and highly threedimensional when all projections are recorded at the same instant. In order to achieve this two tomographic systems have been developed, described more fully in Refs. [7,9]. To solve the problem of ®nding the original distribution (the `source'), the 3D ®eld is usually thought of as being subdivided into a set of parallel planes. Each plane then represents a two-dimensional source ®eld. Using tomographic reconstruction a 2D cross-sectional image (tomoB=slice) of a source function, f(x, z ), is then obtained from its projections, where here we assume that slices are taken at constant values of y. These projections are described by the Radon transform R(t, y ) [10,11]: R t, y f x s, z s ds f x, zd x ÿ t cos y ÿ s sin y, z ÿ s
2
cos y t sin y ds
3 It is also possible to use direct three-dimensional reconstruction (e.g. Ref. [10]), but the equations for this generally are more complicated.
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which in the case of interferometry represents the modi®ed phase of a light beam. Here d is the Dirac delta-function, t is the transverse co-ordinate, y is the viewing angle and s is the co-ordinate along the ray path (see Fig. 1). The integration is idealized as a line integration instead of the actual physical strip integration (which is due to the ®nite thickness of sampling rays). By inverting this equation the refractive index distribution (incorporated in the function f ) and hence the density in the investigated plane can be found. By stacking the reconstructed slices for all parallel planes, the three-dimensional density distribution is obtained3. For continuously sampled data (in both t and y ), R can be inverted directly to obtain the source function. In this case direct analytical inversion using convolution backprojection in signal space (e.g. Ref. [10]) or frequency-domain techniques using ®ltered backprojection (e.g. Ref. [12]) may be used. However, for data sampled discretely in a limited number of views, the results produced by analytical inversion techniques will be contaminated by reconstruction artifacts, which can become quite severe especially when the views are unevenly sampled. Apart from the analytical methods there are three other strategies which may be used for tomographic reconstruction. These three methods have been developed for reconstructing a 3D image from its projections in cases where only limited data is available: approximate analytical inversion (e.g. Ref. [11]), algebraic methods (e.g. ART, MART, SIRT, conjugate gradient) [13±15] and statistical estimation (e.g. [16]). These iterative methods are well-suited for including a priori knowledge and for cases where the angle of sight is limited, as is generally the case in ¯uid dynamics studies. In limited-data tomography using algebraic reconstruction techniques, the reconstructions are represented by a truncated basis function expansion and the projection operation is represented as a matrix, so that the imaging problem is formulated as the system of simultaneous equations, p Hf e
3
where p is the vector of projection measurements, f is the vector of weights of the basis functions, e is the vector of projection measurement errors and H is the projection matrix. The basis function expansion used in these equations may be either local or global. For the reconstructions presented, global basis functions are used, based on Fourier±Bessel expansions. This Fourier±Bessel technique is discussed in two papers by Watt [17,18]. There are two obvious advantages to this expansion. First, since the radial and angular components are represented by separate (but coupled) harmonic expansions, it is easier to match the reconstruction geometry
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to the sampling of the viewing geometry. Second, this expansion easily accommodates densities above and below the ambient, which is common in compressible ¯ows. This global method was found to perform better in cases where views are incomplete (blocked views due to opaque objects) and in cases where much noise is present [17,18]. When the reconstruction grid is generated using Fourier±Bessel functions, points on either side of the object become uncoupled, so that the eect of the blockage is changed. The use of circular harmonics for tomography as proposed by Cormack and Hansen [19±21] has been found to be very convenient in numerous tomographic applications such as in measurements of soft X-ray emissions in Tokamak plasmas [22], which have a circular domain of de®nition. As the jet ¯ows studied here also have a roughly circular symmetry, this approach also seems attractive for this application. The image is reconstructed on a circular domain where the views intersect one another; this is done because the ¯ow is assumed to be contained entirely within this circle. The reconstructed image fà is approximated by a Fourier±Bessel series fà r, y M X K X
Jm amk ramk cos my bmk sin my
4
m0 k0
Here amk are the zeros of the order m Bessel function, r is the normalized radial coordinate and y is the azimuth angle. The Fourier±Bessel expansion has the advantage of imposing a value of zero on the boundaries of the disk, which is an inherent constraint in many reconstructions. The projections are given in terms of the transverse coordinate s and the projection angle f. The coecients of the projection matrix H are calculated by numerical integration of the basis functions Jm(amkr )cos(my ) and Jm(amkr )sin(my ) across the disk along the line of sight of each projection. The resulting algebraic formulation in Eq. (3) expresses the projections p in terms of the unknown f and e. The equations are solved using an iterative conjugate gradient method. This method was adapted following Medo [15] and has been discussed previously by Nakamura [23] and Watt and Conery [24]. The values of M and K are selected to satisfy the Nyquist criteria (or some approximant thereof). The maximum value of M (number of harmonics in azimuthal direction), is basically determined by the number and distribution of the views. In order to resolve the mth spatial harmonic of the source, the mth spatial harmonic of the projection is needed (Eq. 4). For every value of the distance of the projection to the center of the view, there are 2N (N: number of views)
Fig. 2. Vorticity map of compressible ¯ow in a combustion engine based on velocity vector ®eld obtained from PIV measurements. Sizes in mm. (Original in color.)
values for the impact angles available. Using the Nyquist theorem, this is the maximum number of harmonics so that the maximum spatial harmonic that can be resolved is m=N ÿ 1 and one component of m=N (namely the sin(Nc ) or the cos(Nc ) component). Any higher harmonics cannot be resolved and will be aliased down to lower spatial harmonics. The maximum value of K is the number of zeroes and the number of cycles over the radius. Therefore, for example, K = 23 produces 92 oscillations over the diameter.
3. Turbulent structures Generally in turbulence studies coherent structures are de®ned as structures having a constant vorticity, which may be identi®ed based on velocity measurements. An example of this is shown in Fig. 2 where PIV results are shown which were obtained in a compressible combustion ¯ow. In Fig. 2, the measured vorticity ®eld for the ¯ow between the compressor blades is shown as a grey value image (original in color). In this image clearly areas may be discerned which have a de®nitely dierent vorticity value compared to their surroundings. The term `structure' may be used to describe coherent, organized motions occurring in the ¯ow. An often
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Fig. 3. (a) Source image. (b) Reconstructed image (rms error 0.17). Simulated source image and reconstruction of source image from projections at ÿ12, 0, 12, 78, 90 and 1028 using the same Fourier±Bessel expansion as in experiments. (M = 4, K = 24).
used de®nition in this sense was given by Robinson [25], who de®ned a coherent motion or structure as [26] a three-dimensional region of the ¯ow over which at least one fundamental ¯ow variable (velocity component, density, temperature, etc.) exhibits signi®cant correlation with itself or with another ¯ow variable over a range of space and/or time that is signi®cantly larger than the smallest local scales of the ¯ow. Organized structures are found to exist in several turbulent ¯ows and are responsible for turbulence production and transportation. A better understanding of the dynamics of these structures is a major issue for
the improvement of turbulence modeling and the implementation of eective turbulence control. The study of large scale turbulent structures is thus important because they appear to dominate several turbulent ¯ows.
4. Results HIT allows the study of both absolute as well as dierential turbulent density structures [7]. The resolution that may be obtained for determining these structures, depends on the experimental viewing con®guration that is used to obtain the data, as described in the previous section. The resolution of the tomo-
Fig. 4. Sinogram for underexpanded, supersonic case of the 6 absolute phase maps. Viewing order from left to right: 78, 90, 102, ÿ12, 0 and 128, viewing area: 7 14.4 mm2 (101 207 pixels) per view starting immediately below the nozzle exit.
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Fig. 5. Iso-surface at a density of 1.35 kg/m3 for the supersonic case. Volume: 5 5 5 mm3, starting immediately at the nozzle exit, ¯ow direction: top-down.
graphic system has been evaluated theoretically for the speci®c tomography con®gurations used in this experiment using a numerically generated test ®eld. The test ®eld has also been used in previous studies by Watt [17,18,24]. It represents a turbulent ¯ow ®eld, containing both the smooth and ®ne scale features (Fig. 3a). Fig. 3(b) shows the reconstruction of the simulated turbulent ®eld using 6 views, which are unevenly spread in two orthogonal fans, creating projections at ÿ12, 0, 12, 78, 90 and 1028. This reconstruction was made using 4 harmonics in azimuthal direction and 24 in radial direction, approximating the Nyquist criteria for a 6 view con®guration with 101 pixels in each view. As can be seen from this reconstruction, the ®ne scaled features present in the source image are not reproduced. However, the large scale overall features are found even using this low number of views. To test what kind of results may be obtained in actual experiments some test ¯ows were generated using a 450 mm long cylindrical tube, with an internal diameter of 3 mm. At suciently high pressures at the nozzle exit, expansion to supersonic conditions may be expected to result in the well-known diamond-shaped pattern for the density distribution in the resulting free jet, which will decay due to free shear layer inter-
actions. At the supersonic condition studied here the pipe produced an underexpanded jet ¯ow which is stable when leaving the pipe but becomes unsteady at about 2.5 diameters from the nozzle exit as can be seen in Fig. 4 where 6 phase maps obtained from dierent viewing directions, corresponding to those used to generate the simulated result shown in Fig. 3(b), are shown in the form of a so-called sinogram. These six absolute phase maps were made at the same instant of time, thus providing information on the instantaneous behavior of the ¯ow ®eld. The individual phase images were unwrapped, corrected for tilt and magni®cation to obtain the projection measurements at 101 points in each of a number of planes centered in the nozzle. These corrected projections were then collected into the sinogram, which contains the projections obtained from a single volume. Each plane in the sinogram was then reconstructed using the Fourier± Bessel technique described in Section 3. Each image in the sinogram image of Fig. 4 is actually produced from two 512 512 images grabbed by the CCD camera from the same hologram in order to improve the resolution. Thus, smaller features may be resolved while retaining a large ®eld of view, in this case especially in the direction of the ¯ow.
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Fig. 6. (a) Iso-surface at a density of 1.47 kg/m3 (also shown are markers that were put in after the reconstruction). (b) Slice through y, z-centerplane of reconstruction. Tomographic reconstruction of free jet ¯ow using 6 view set-up. Volume: 5 5 5 mm3, starting immediately at nozzle exit; ¯ow direction: top-down.
The phase values are rendered in grey tones, where the (zero-phase-dierence) background is rendered in a light shade of grey and the phase dierence increases from grey through white to black. The maximum phase dierence in each view is about 1.26 2p, corresponding to a maximum average density inside the jet of 1.29 kg/m3 above ambient (with l=693.4 nm and based on a ¯ow width of 3 mm). In the phase maps of Fig. 4, going from the nozzle along the centerline, ®rst a decreasing phase can be seen agreeing with an expansion in the nozzle exit. Further downstream the phase increases again. After re¯ection of the shock waves at the free jet boundary, subsequent expansion causes the (integrated) density to decrease again. This repeated compression±expansion sequence results in the familiar diamond shaped pattern for the integrated density in the steady part of the ¯ow ®eld. After some streamwise distance the ¯ow develops an instability, which is re¯ected by the wavy pattern in the lower part of the phase map of Fig. 4. Fig. 5 shows an iso-density-surface of the 3D reconstructions for the upper part of this volume at a density level of 0.15 kg/m3 with respect to ambient. For this reconstruction 101 101 pixels were used per viewing direction, representing a volume of 5 5 5 mm3. In Fig. 5, from the shape of the iso-density-surface, two compression areas can be recognized, one starting immediately at the nozzle exit, the other somewhat further downstream (cf. Fig. 4). Also an increasing complexity of the ¯ow can be seen.
As a further example of the results that may be obtained in actual experiments, in Fig. 6 a tomographic reconstruction is shown of a typical subsonic free jet using the viewing con®guration of Fig. 3(b) and the same ¯ow generator as used for the previous results. Again all six absolute phase maps used in the reconstruction were recorded at the same instant on the holographic plate. Prior to reconstruction, the projections in the sinogram were ®ltered with a twodimensional lowpass ®lter. For the reconstruction shown here, 101 101 pixels were used per view, which in this case represents an area of 5 5 mm2 starting immediately below the nozzle exit. Fig. 6(a) shows an iso-density surface, while Fig. 6(b) shows the center slice through the ¯ow ®eld. The iso-density surface shows that the jet-shape evolves from a basically circular form close to the pipe exit, into a noncircular shape starting about one pipe-diameter from the exit. This spatially non-uniform character of the ¯ow is seen more clearly from the center slice through the jet shown in Fig. 6(b). Fig. 6(b) shows the density values in the form of grey scales. Thus, several light shaded connected areas can be seen, representing dierent density structures that are present in the ¯ow. From Fig.6(b) an increase in structure size and strength can be seen with increasing distance from the pipe exit, indicating an increasing turbulence level. A series of these slices, centered around the volume axis in the x, z-plane is shown in Fig. 7. These slices re¯ect the 3D character of the recorded ¯ow ®eld. Following the structures in Fig. 7 from (a) to (e) it can
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Fig. 7. (a) Slice at ÿ0.7 mm from center of nozzle; (b) slice at ÿ0.2 mm from center of nozzle; (c) slice at +0.2 mm from center of nozzle; (d) slice at +0.7 mm from center of nozzle and (e) slice at +1.2 mm from center of nozzle. Slices around the center of studied volume, taken in the x, z-plane (originals in color).
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Fig. 8. (a) Source image. (b) Reconstructed image (rms error: 0.09). Simulated source image and reconstruction of source image from projections at ÿ70.2, ÿ47.1, ÿ31.3, ÿ16.9, 0, 16.9, 30.8, 47.1 and 69.78, using the same Fourier±Bessel expansion as in experiments (M = 9, K = 23).
be seen that in (a) the structures are mostly present on the top left side, near the nozzle exit, while at position (e) structures are seen on the lower right side furthest from the nozzle. Furthermore, it becomes clear that the actual ¯ow is not centered around the axis of the nozzle in this plane, but appears to be shifted by
about 0.2 mm, which also appears to be the case for the supersonic ¯ow discussed earlier. This 6 view con®guration is obviously not ideal, as may be judged directly from the simulation results shown in Fig. 3. The number of views is limited and also the angular spread is limited, resulting in sparse
Fig. 9. (a) Iso-density surface showing columns connecting compression regions, density level: 0.32 kg/m3 with respect to ambient. (b) Axial slice through edge of jet; x, z-plane. Tomographic reconstruction of underexpanded free jet ¯ow, based on 9 views. ps=3 bar, pipe inner diameter d = 6.5 mm, volume: 20.4 20.4 20.5 mm3. Flow direction: top-down.
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Fig. 10. Tomographic results for oscillating free jet ¯ow showing iso-density-dierence surfaces. Bright regions: density dierence of 0.14 kg/m3, dark regions: ÿ0.14 kg/m3. D T =76 ms, Flow direction: top to bottom, volume 17 17 16.88 mm3.
sampling of the investigated ®elds. Furthermore, this con®guration also provides an uneven sampling, reducing the obtainable accuracy even further. To improve the quality of the reconstructions that may be obtained, a tomographic system has been developed, which records 9 simultaneous views, again at the same instant of time [9]. The beams produced in this tomographic set-up simultaneously traverse the test ®eld at angles of ÿ70.2, ÿ47.1, ÿ31.3, ÿ16.9, 0, 16.9, 30.8, 47.1 and 69.708, producing a tomographic system which is symmetrical with respect to the z-axis (direction of original object beam). This con®guration provides views, which are roughly equally distributed over a range of about 140 degrees. The ®eld of view of the present set-up is about 40 40 40 mm3. Fig. 8 shows the results that may be obtained using the Fourier±Bessel technique for the simulated turbulent ®eld for this 9-view con®guration. In this case 9 harmonics were used in azimuthal direction and 23 in the radial direction. The resolution that may be obtained using this con®guration is clearly greatly improved, as now also smaller scale structures appear to be resolved. From Fig. 8 it can be seen that the size of the structures that can be discerned using this setup is about 1/50 of the original projection size used for the reconstruction, when the object covers about 40%
of the projection size. In that case, therefore, the size of the structures that can be resolved potentially is about 1/20th of the size of the original ®eld. In the cases shown here this corresponds to about 250 mm. Fig. 9 shows a tomographic reconstruction of an underexpanded free jet ¯ow. This ¯ow is generated by a con®guration consisting of a high pressure line which supplies a settling chamber with an inner diameter of 35 mm, which contracts smoothly into a round tube which is 60 mm long. Thus, the studied jet ¯ow issues from a straight pipe with a 6.5 mm inner diameter. The tomographic reconstruction shows the ¯ow to be essentially circular in the diamond shaped compression regions, whereas the areas connecting the compression regions show some clearly three-dimensional features (Fig. 9a). These ¯ow features are as would be expected for a ¯ow from a (relatively long) round pipe. Because of the pipe length, a turbulent ¯ow will be generated, which may contain irregular patterns even close to the exit, especially in the shear layer regions. Fig. 9(b) shows an axial slice through the outer layer of the jet at about 3.2 mm from the center plane. This image clearly indicates the unsteady, oscillatory character of the ¯ow ®eld, which increases in strength as it progresses downstream. A comparison of the tomographic results obtained for this ¯ow to
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results obtained for the same ¯ow using an experimental Rayleigh scattering technique and a numerical Euler technique show an encouraging similarity [27]. As the Euler technique only produces steady ¯ow results and the Rayleigh results were averaged over several laser shots, some of the dierences that were found may be due to the actual unsteady character of the ¯ow, which is only captured by the tomographic technique. An indication of this unsteady behavior was also found by Dam et al. [28], in their Rayleigh scattering measurements when comparing images obtained from single laser pulses. The density areas that may be identi®ed in the tomographic results such as shown in Fig. 7 may possibly be interpreted as coherent density structures (cf. Fig. 2). To positively identify these as coherent structures however, their behavior in time should also be followed. By recording dierential interferograms using the rapid-switching facility of the DHI set-up [7], it is possible to measure the density changes that occur in the ¯ow over an adjustable time-interval. When coherent density structures are imagined to be density volumes travelling through the ¯ow [29], the resulting dierential density results may be interpreted as follows: When a (single) structure moves in an otherwise homogeneous area (idealized example, for time intervals which are long enough for the structure to have moved a distance greater than its own length), the phase map consists of two structure images, displaced in the ¯ow direction. One has a decreased phase and the other an increased phase, more or less similar in form, assuming that the structures are only displaced and not deformed in the time interval. As an illustration of the capabilities of the combination of the 9-view tomographic method and the quantitative double-pulsed interferometry, Fig. 10 shows a tomographic reconstruction of a dierential measurement of a free jet ¯ow that is oscillating at a frequency of 6550 Hz [30]. In Fig. 10, a dierential recording at a pulse-interval of half the oscillation period is shown. The dierential recording shows volumes with increased and decreased density. This may be interpreted as volumes between which the structures have moved. In the case shown here, it is seen that structures are displaced vertically as well as in-plane. Thus, it is shown that the tomographic doublepulsed technique may be used in order to localize density structures and to determine and investigate their shape, strength and velocities (at least for the larger structures). In the case shown here, the velocity with which the structures have moved may be estimated roughly to be in the order of 70 m/s.
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5. Discussion The actual resolution that has been obtained using the 9 view tomography system is about 1 mm in size while the density level that may be discerned is about 0.04 kg/m3 [27]. The resolution of this system is limited by the number and spread of the views and also by the number of projections per view. Generally, it is found that the improvement in reconstruction accuracy decreases with increasing these parameters [31,32]. Therefore, increasing these beyond a certain limit barely improves the reconstruction as the additional projections are not suciently independent from the existing ones to provide non-redundant information. For a simulated turbulent ®eld, containing no shocks nor opaque objects, Watt and Conery [24] found that no signi®cant improvement of the reconstruction will be obtained by using more than 12 views. Holographic data has been used in this paper to tomographically reconstruct and visualize the three dimensional turbulent structure within a transonic jet. The structure is of interest because of the detail which has been obtained. Having used holographic interferometry and tomography to de®ne the three-dimensional instantaneous turbulent structure within the transonic ¯ow, it is of interest to explore both its spatial and temporal behavior. The tomographic system described in this paper was designed for application, with minimum alteration, to the high speed wind tunnels at Delft. To achieve this the system needed to be both compact and rigid. With the advent of high resolution high speed digital cameras it is now possible to replace the holographic recording medium. In this way a real-time interferometric system may be created where for each viewing direction the phase-stepped phase maps are simultaneously recorded. A further simpli®cation is possible by the use of a shearing interferometer [33]. Such systems are far less sensitive to vibration or mechanical path changes and are optically very simple to build. Also, although the measured projections may be more noisy in the case of shearography than for interferometry, recently Watt [18] showed, based on numerical simulations, that gradient type reconstructions are no more susceptible to measurement noise than projection techniques. The holographic interferometry (HI) described in this paper is not real time. However, by using either of the approaches described by Pellicia and Watt [34] or Fomin et al. [35], combined with the knowledge of the structure obtained from HI, it is possible to see how a real time system could be developed. Pellicia and Watt replace HI with shearography, which uses Fourier shift theory to unwrap the phase maps, prior to its tomographic reconstruction. This approach is an order of magnitude less sensitive than phase stepping but does
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allow a high speed capture approach to the problem and also a fully automated data extraction. Shearography also provides a dierential measurement, which requires some interpretation with respect to the turbulent structures. Its main diculty however, is the experimental complexity of the system, which needs to extract multiple views. This restricts its application to open ¯ows, which do not have a viewing limitation. An alternative, statistical, approach is proposed by Fomin et al. [35]. They use the refractive index ray bending which occurs as the light passes through a turbulent structure to de®ne its position in the direction of light path propagation. To achieve this an auxiliary measurement is used, which is based on the average size distribution of the turbulent structure. This is then used to model the ¯ow structure and then match the positional bending of the ray by calculation. The refractive index bending is a source of error in the HI case and is ignored as being much smaller than the ray path phase change through the structure. However, in the current case HI has been used to de®ne the size and shape of the turbulent structures in three dimensions. In particular it has shown the amount of structure within the jet is small and relatively well de®ned. Thus, by using this data, a model of shapes and structures could be used to increase the accuracy of the statistical method. The advantage being that once validated in this manner, the experiment only requires one direct view through the ¯ow ®eld. This opens up the possibility of extracting both the temporal and three dimensional spatial evolution of complex turbulent high speed structures in a greater detail than previously obtained. 6. Conclusions The results presented here indicate that the tomographic interferometry technique is able to provide quantitative information regarding the three-dimensional structure of real, unsteady ¯ows. Large-scale, unsteady density structures may be identi®ed, at density-dierences down to 0.04 kg/m3, showing the complete three-dimensional temporal behavior of ¯ows. Details of the spatial structural evolution of transonic free jet ¯ows have been obtained. The results show an increase in the size and complexity of this structure with distance along the jet. The result also presents a visualization of the oscillatory unsteady ¯ow nature in the outer layers of the jet. By changing the pulse-interval in double-pulse experiments, unsteady or oscillating ¯ows may be followed in time. This gives insight into the behavior of unsteady ¯ows. A single instantaneous image by itself does not provide information on the temporal be-
havior, whereas dierential measurements only show the changes in the ¯ow. Combined, however, they oer a understanding of both the spatial and temporal ¯ow behavior. Furthermore, from dierential measurements, coherent structures may not only be identi®ed, but also their velocity can be determined. Thus, the speed of mixing and transportation phenomena may be studied. Also using this technique, it is possible to extract turbulence time-scale information from dierential interferometric results for an arbitrary compressible ¯ow ®eld. Most importantly, the tomographic technique has provided an instantaneous three-dimensional visualization of both the steady and unsteady behavior of the ¯ow which could not have been achieved using more conventional techniques.
Acknowledgements The technical support of Frits Donker-Duyvis and Eric de Keizer of the Laboratory for High Speed Aerodynamics of the TU Delft is greatly appreciated. DWW's contributions to this project were supported by NSF Grant Number ECS8910350 and the NATO Scienti®c Research Programme. Part of BHT's work was funded by the Dutch Organisation for Scienti®c Research (NWO) through Projectnumber DLR33.3109 of the Technology Foundation (STW).
References [1] Bryanston-Cross PJ, Judge TR, Quan C, Pugh G, Corby N. The application of digital particle image velocimetry (DPIV) to transonic ¯ows. Prog Aerospace Sci 1995;31:273±90. [2] Meyers JF. Development of Doppler global velocimetry as a ¯ow diagnostics tool. Measurements in ¯uids and combustion systems [special issue]. Meas Sci Technol 1995;6(6):769±83. [3] Lanen TAWM, Bakker PG, Bryanston-Cross PJ. Digital holographic interferometry in high-speed ¯ow research. Exp Fluids 1992;13:56±62. [4] Merzkirch W. Flow visualization, 2nd ed. Academic Press, 1987. [5] Vest CM. Holographic interferometry. Wiley, 1979. [6] Lanen TAWM. Digital holographic interferometry in compressible ¯ow research. Ph.D. thesis, University of Technology, Delft, October 1992. [7] Timmerman BH, Watt DW. Tomographic high-speed digital holographic interferometry. Meas Sci Technol 1995;6(9):1270±7. [8] Bone DJ. Fourier fringe analysis: the two-dimensional phase unwrapping problem. Appl Opt 1991;30(25):3627±32. [9] Timmerman BH, Watt DW. Tomographic interferometry for unsteady compressible ¯ows. In: Optical techniques in ¯uid, thermal and combustion ¯ow, Vol. 2546. San Diego, CA: SPIE, 1995. p. 287±96. [10] Deans SR. The radon transform and some of its applications. NY: Wiley and Sons, 1983. [11] Watt DW, Vest CM. Turbulent ¯ow visualization by interferometric integral imaging and computed tomography. Exp Fluids 1990;8:301±11.
B.H. Timmerman et al. / Optics & Laser Technology 31 (1999) 53±65 [12] Rosenfeld A, Kak AC. Digital picture processing. I. Academic Press, 1982. [13] Gilbert P. Iterative methods for the three-dimensional reconstruction of an object from projections. J Theor Biol 1972;36:105±17. [14] Herman GT, Lent A. Iterative reconstruction algorithm. Comput Biol Med 1976;6:273±94. [15] Medo BP. Image reconstruction from limited data: theory and application in computerized tomography [ch. 9]. In: Stark H, editor. Image recovery: theory and application. NY: Academic Press, 1987. p. 321±68. [16] Levitan E, Herman GT. A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography. IEEE Trans Med Imaging 1987;MI6(3):185±92. [17] Watt DW. Fourier±Bessel harmonic expansions for tomography of partially opaque objects. Appl Opt 1995;34:7468±73. [18] Watt DW. Fourier±Bessel expansions for tomographic reconstruction from phase and phase-gradient measurements: theory and experiment. In: Symposium on Experimental and Numerical Flow Visualization and Laser Anemometry, ASME Fluids Engineering Division Summer Meeting, FEDSM-3102, June 22±26, Vancouver, BC, Canada, 1997. [19] Cormack AM. Representation of a function by its line integrals, with some radiological applications. J Appl Phys 1963;34(9):2722±7. [20] Cormack AM. Representation of a function by its line integrals, with some radiological applications II. J Appl Phys 1964;35(10):2908±13. [21] Hansen EW. Theory of circular harmonic image reconstruction. J Opt Soc Am 1981;71(3):304±8. [22] da Cruz DF, Donne AJH. Soft X-ray tomography diagnostic at the RTP Tokamak. Rev Sci Instrum 1994;65(7):2295±305. [23] Nakamura O, Kawata S, Minami S. Optical microscopic tomography. II Non-negative constraint by a gradient-projection method. J Opt Soc Am A 1988;5(4):554±61. [24] Watt DW. Computational aspects of tomographic reconstruction for ¯uid mechanics, heat transfer and combustion. In: Third ASME Symposium on Experimental and Numerical Flow Visualization, December 1±4, New Orleans, LA, USA, 1993. [25] Robinson SK. Coherent motions in the turbulent boundary layer. Annu Rev Fluid Mech 1991;23:601±39.
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[26] Smits AJ, Dussauge JP. Turbulent shear layers in supersonic ¯ow. AIP Press, 1996. [27] Timmerman BH. Holographic interferometric tomography for unsteady compressible ¯ows. Ph.D. thesis, University of Technology, Delft. Eburon P&L, November 1997. [28] Dam NJ, Rodenburg M, Tolboom RAL, Stoels GGM, Huisman-Kleinherenbrink PM, ter Meulen JJ. Imaging of an underexpanded nozzle ¯ow by UV laser Raleigh scattering. Exp Fluids 1998;24(2):93±101. [29] Zhong S, Squire LC. Visualisation of organized structures in compressible turbulent ¯ow by holographic interferometry. In: Adrian RJ, DuraÄo DFG, Durst F, Heitor MV, Maeda M, Whitelaw JH, editors. Sixth International Symposium on the Application of Laser Techniques to Fluid Mechanics. Lisbon, Portugal: Springer Verlag, 1993. p. 433±50. [30] Timmerman BH, Watt DW. Tomographic holographic interferometric study of self-excited compressible jet ¯ows. In: Kawahashi M, Cha SS, Trolinger JD, editors. Optical technology in ¯uid, thermal and combustion ¯ow III, Vol. 3172. San Diego, CA: SPIE, 1997. p. 307±15. [31] McCaughey DG, Andrews HC. Degrees of freedom for projection imaging. IEEE Trans Acoust Speech Signal Proc 1977;ASSP-25(1):63±73. [32] Lanen TAWM, Watt DW. Singular value decomposition: a diagnostic tool for ill-posed inverse problems in optical computed tomography. In: Experimental and numerical methods for solving ill-posed inverse problems: medical and nonmedical applications, Vol. 2570. San Diego, CA: SPIE, 1995. p. 174±85. [33] Burnett M, Bryanston-Cross PJ. Measurements of transonic shock structures using shearography. In: Pryputniewicz RJ, Brown GM, Jueptner WP, editors. Laser interferometry VIII: applications, Vol. 2861. Denver, CO: SPIE, 1996. p. 124±35. [34] Pelliccia BJ, Watt DW. Three-dimensional imaging of turbulent ¯ows: optical tomography and proper orthogonal decomposition. In: 8th International Symposium on Flow Visualization, Sorrento, Italy, 1998. [35] Fomin N, Lavinskaja E, Merzkirch W, Vitkin D. Statistical analysis of turbulence by speckle photography. In: International Conference on Optical Methods and Data Processing in Heat and Fluid Flow, IMechE Conference Transaction 1998-2, 16±17 April, London, UK, 1998. p. 433±42.
www.elsevier.com/locate/optlastec
Quantitative visualization of high-speed 3D turbulent ¯ow structures using holographic interferometric tomography B.H. Timmerman a, 1, D.W. Watt b, 2, P.J. Bryanston-Cross c,* a
Department of Aerospace Engineering, Delft University of Technology, Laboratory for High Speed Aerodynamics, P.O. Box 5058, 2600 GB Delft, Netherlands b Department of Mechanical Engineering, The University of New Hampshire, Kingsbury Hall, Durham, NH 03824, USA c Optical Engineering Laboratory, Department of Engineering, University of Warwick, Coventry CV4 7AL, UK
Abstract Using holographic interferometry the three-dimensional structure of unsteady and large-scale motions within subsonic and transonic turbulent jet ¯ows has been studied. The instantaneous 3D ¯ow structure is obtained by tomographic reconstruction techniques from quantitative phase maps recorded using a rapid-switching, double reference beam, double pulse laser system. The reconstruction of the jets studied here reveal a three-dimensional nature of the ¯ow. In particular an increasing complexity can be seen in the turbulence as the ¯ow progresses from the jet nozzle. Furthermore, a coherent three-dimensional, possibly rotating, structure can be seen to exist within these jets. The type of ¯ow features illustrated here are not just of fundamental importance for understanding the behavior of free jet ¯ows, but are also common to a number of industrial applications, ranging from the combustion ¯ow within an IC engine to the transonic ¯ow through the stages of a gas turbine. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Tomography; Interferometry; Turbulent structures
1. Introduction The number of quantitative ¯ow investigation tools available in engineering has increased rapidly over recent years thanks to fast developing technologies. This has made it possible for some techniques, such as laser Doppler and Laser-2-focus, to mature from being laboratory research instruments to becoming commonplace industrial tools. However, although these techniques provide quantitative measurements, they remain essentially point measurement devices. Thus, they require signi®cant data acquisition times to map a whole ¯ow ®eld which makes them a poor tool for * Corresponding author. Tel.: +44-1203-523-131; fax: +44-1203418-922. 1 Tel.: +31-15-278-5349; fax: +31-15-278-7077. E-mail address: B. [email protected] (B.H. Timmerman) 2 Tel.: +1-603-862-25555; fax: +1-603-862-2486.
globally mapping such features as the turbulent structure within an unsteady transonic ¯ow. To overcome this limitation whole ®eld instruments like PIV [1] and global Doppler velocimetry [2] are emerging as industrial tools. However, these methods are still both highly dependent on the need to `seed' the ¯ow ®eld with particles. In several cases the nature of the seeding may be a prohibitive limitation. This is the case for instance when there are ¯uctuations in a ¯ow which are in the high kHz range, like those found in structures within turbulent transonic jet ¯ows. For the seeding to follow the ¯ow accurately in these cases, the particles need to be smaller than 0.2 mm in size. Bigger particles may then give erroneous information. This is also important when `following' a travelling shock wave, which again demands a very fast inertial response of the seeding particulate. Also, the nature of the ¯ow studied may inhibit the use of seeding. Examples of this are the highly volatile regions such as found in combustion ¯ows, where the life-time of a
0030-3992/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 9 ) 0 0 0 3 6 - 5
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small particle is `short' or ¯ows where there is a concentration of other larger particle debris in the ¯ow. Furthermore, in some cases the introduction of a particulate can create a signi®cant perturbation to the ¯ow ®eld. In these cases there is a need for a technique that requires no seeding particles to make a measurement of the ¯ow. Digital holographic interferometry (DHI) [3] uses properties inherent to the ¯ow, like temperature or density dierences, thus enabling a nonintrusive, instantaneous study of such compressible ¯ows. The drawback to interferometric techniques is that they provide line-of-sight measurements. Because the majority of compressible ¯ow features of interest are three-dimensional, a method is required to extract local information from the projected data. The holographic interferometric tomography (HIT) method described here uses a number of multiple views which are formed simultaneously and stored holographically to tomographically reconstruct this local 3D data. Thus, both the large and smaller scale turbulence structures of a transonic jet ¯ow may be visualized in three dimensions. Over recent years many attempts have been made to extract three-dimensional data using interferometric tomography. As the method is numerically intensive and generally uses underde®ned data it has tended to yield low resolution and uncertain representations of the ¯ow ®eld. By combining the quantitative accuracy, as is obtained from automatically phase unwrapped interferometric images and a higher resolution stable tomographic solution this problem has been overcome. This paper shows how a high resolution tomographic solution can provide a detailed description of the complex multi scale structure and its spatial development found within a transonic jet ¯ow. Several speci®c features, like the three-dimensional coherent structure of the jet have been visualized using the HIT process which would have not been possible using other techniques. Also the structure of the turbulence and its spatial development can be seen. Further by making dierential interferograms at dierent time intervals the timescale, shape, strength and velocity of the structures can be determined. Such observations may be very relevant in studying the mixing rate and eciency in the combustion process and stage ef®ciency in turbomachinery. Furthermore to further develop numerical three-dimensional unsteady transonic ¯ow simulation models, calculations obtained by these models have to be evaluated against the type of data shown in this paper. Thus, a better understanding of the underlying processes may be obtained, which in turn creates new and more ecient power generation systems. The interferometric set-up presented performs an in-
stantaneous and simultaneous recording of multiple projections at dierent angles. From these projections the instantaneous 3D density (¯uctuation) ®eld can be obtained using tomographic reconstruction techniques. The performance of the HIT system was assessed for several types of compressible free jet ¯ows.
2. Holographic interferometric tomography 2.1. Holographic interferometry Holographic interferometry is a technique that may be used to study the refractive index distribution in transparent media. By sending interrogation beams through the region of interest optical ray path integrals, representing the refractive index ®eld are obtained in the form of phase maps. For compressible ¯ow ®elds the refractive index is related to density by the Gladstone±Dale relation [4]. Ignoring refraction, this relation can be used to relate the gas density directly to a phase change of the light ray passing through it [5]: 2pK r x, y, z ÿ r0 dz 1 Df x, y l where l is the wavelength of the probe beam, r is the density distribution, r0 is the (constant) background/ reference density ®eld, Df is the phase dierence between a beam passing through the ¯ow ®eld and one which only passes the background ®eld and K is the Gladstone±Dale constant, which is about 0.225 103 m3/kg for air at a probing wavelength of 694 nm. The phase data used in this paper is obtained from a two-reference beam, plane wave, tomographic holographic interferometer which uses a ruby pulse laser to holographically store the phase distribution associated with the ¯ow ®eld [3,6]. Unsteady ¯ows may be studied by operating the laser in the `Q' switched mode, giving 30 ns duration laser pulses which eectively freeze the motion of the ¯uid even at high ¯uctuation rates. The system allows separate recording of two ¯ow conditions at an adjustable microseconds timeinterval using the laser in double-pulsed mode, replaying of the holograms in exactly the same set-up using a HeNe laser and spatial ®ltering to remove undesirable cross-reconstructions. Thus, errors due to optics or badly repositioning of the hologram are avoided while speckle-free, plane-wave interferograms are generated which are stored using a CCD camera (512 512 pixels) [7]. Both the double exposure and double-pulsed interferograms are processed using phase-stepping techniques [6]. The 2p phase jumps in the wrapped phasemaps are then removed using phase-unwrapping [8] to
B.H. Timmerman et al. / Optics & Laser Technology 31 (1999) 53±65
Fig. 1. Projection-imaging geometry.
obtain the continuous phase data at an accuracy of approximately 1/10th of a fringe [6]. The double exposure method allows a quantitative measurement of both the instantaneous compressible ¯ow and the rapidly changing unsteady density ¯uctuations [7]. 2.2. Tomography The purpose of optical tomography is to obtain the 3D distribution of a scalar quantity from a set of projections like those obtained in compressible ¯ow interferometry. There, spatially resolved 3D density information can be obtained by tomographically reconstructing the data from a series of ray path projections along dierent viewing angles through a ¯ow ®eld. HIT, for example, can be used to measure turbulent ¯ows, which are both unsteady and highly threedimensional when all projections are recorded at the same instant. In order to achieve this two tomographic systems have been developed, described more fully in Refs. [7,9]. To solve the problem of ®nding the original distribution (the `source'), the 3D ®eld is usually thought of as being subdivided into a set of parallel planes. Each plane then represents a two-dimensional source ®eld. Using tomographic reconstruction a 2D cross-sectional image (tomoB=slice) of a source function, f(x, z ), is then obtained from its projections, where here we assume that slices are taken at constant values of y. These projections are described by the Radon transform R(t, y ) [10,11]: R t, y f x s, z s ds f x, zd x ÿ t cos y ÿ s sin y, z ÿ s
2
cos y t sin y ds
3 It is also possible to use direct three-dimensional reconstruction (e.g. Ref. [10]), but the equations for this generally are more complicated.
55
which in the case of interferometry represents the modi®ed phase of a light beam. Here d is the Dirac delta-function, t is the transverse co-ordinate, y is the viewing angle and s is the co-ordinate along the ray path (see Fig. 1). The integration is idealized as a line integration instead of the actual physical strip integration (which is due to the ®nite thickness of sampling rays). By inverting this equation the refractive index distribution (incorporated in the function f ) and hence the density in the investigated plane can be found. By stacking the reconstructed slices for all parallel planes, the three-dimensional density distribution is obtained3. For continuously sampled data (in both t and y ), R can be inverted directly to obtain the source function. In this case direct analytical inversion using convolution backprojection in signal space (e.g. Ref. [10]) or frequency-domain techniques using ®ltered backprojection (e.g. Ref. [12]) may be used. However, for data sampled discretely in a limited number of views, the results produced by analytical inversion techniques will be contaminated by reconstruction artifacts, which can become quite severe especially when the views are unevenly sampled. Apart from the analytical methods there are three other strategies which may be used for tomographic reconstruction. These three methods have been developed for reconstructing a 3D image from its projections in cases where only limited data is available: approximate analytical inversion (e.g. Ref. [11]), algebraic methods (e.g. ART, MART, SIRT, conjugate gradient) [13±15] and statistical estimation (e.g. [16]). These iterative methods are well-suited for including a priori knowledge and for cases where the angle of sight is limited, as is generally the case in ¯uid dynamics studies. In limited-data tomography using algebraic reconstruction techniques, the reconstructions are represented by a truncated basis function expansion and the projection operation is represented as a matrix, so that the imaging problem is formulated as the system of simultaneous equations, p Hf e
3
where p is the vector of projection measurements, f is the vector of weights of the basis functions, e is the vector of projection measurement errors and H is the projection matrix. The basis function expansion used in these equations may be either local or global. For the reconstructions presented, global basis functions are used, based on Fourier±Bessel expansions. This Fourier±Bessel technique is discussed in two papers by Watt [17,18]. There are two obvious advantages to this expansion. First, since the radial and angular components are represented by separate (but coupled) harmonic expansions, it is easier to match the reconstruction geometry
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to the sampling of the viewing geometry. Second, this expansion easily accommodates densities above and below the ambient, which is common in compressible ¯ows. This global method was found to perform better in cases where views are incomplete (blocked views due to opaque objects) and in cases where much noise is present [17,18]. When the reconstruction grid is generated using Fourier±Bessel functions, points on either side of the object become uncoupled, so that the eect of the blockage is changed. The use of circular harmonics for tomography as proposed by Cormack and Hansen [19±21] has been found to be very convenient in numerous tomographic applications such as in measurements of soft X-ray emissions in Tokamak plasmas [22], which have a circular domain of de®nition. As the jet ¯ows studied here also have a roughly circular symmetry, this approach also seems attractive for this application. The image is reconstructed on a circular domain where the views intersect one another; this is done because the ¯ow is assumed to be contained entirely within this circle. The reconstructed image fà is approximated by a Fourier±Bessel series fà r, y M X K X
Jm amk ramk cos my bmk sin my
4
m0 k0
Here amk are the zeros of the order m Bessel function, r is the normalized radial coordinate and y is the azimuth angle. The Fourier±Bessel expansion has the advantage of imposing a value of zero on the boundaries of the disk, which is an inherent constraint in many reconstructions. The projections are given in terms of the transverse coordinate s and the projection angle f. The coecients of the projection matrix H are calculated by numerical integration of the basis functions Jm(amkr )cos(my ) and Jm(amkr )sin(my ) across the disk along the line of sight of each projection. The resulting algebraic formulation in Eq. (3) expresses the projections p in terms of the unknown f and e. The equations are solved using an iterative conjugate gradient method. This method was adapted following Medo [15] and has been discussed previously by Nakamura [23] and Watt and Conery [24]. The values of M and K are selected to satisfy the Nyquist criteria (or some approximant thereof). The maximum value of M (number of harmonics in azimuthal direction), is basically determined by the number and distribution of the views. In order to resolve the mth spatial harmonic of the source, the mth spatial harmonic of the projection is needed (Eq. 4). For every value of the distance of the projection to the center of the view, there are 2N (N: number of views)
Fig. 2. Vorticity map of compressible ¯ow in a combustion engine based on velocity vector ®eld obtained from PIV measurements. Sizes in mm. (Original in color.)
values for the impact angles available. Using the Nyquist theorem, this is the maximum number of harmonics so that the maximum spatial harmonic that can be resolved is m=N ÿ 1 and one component of m=N (namely the sin(Nc ) or the cos(Nc ) component). Any higher harmonics cannot be resolved and will be aliased down to lower spatial harmonics. The maximum value of K is the number of zeroes and the number of cycles over the radius. Therefore, for example, K = 23 produces 92 oscillations over the diameter.
3. Turbulent structures Generally in turbulence studies coherent structures are de®ned as structures having a constant vorticity, which may be identi®ed based on velocity measurements. An example of this is shown in Fig. 2 where PIV results are shown which were obtained in a compressible combustion ¯ow. In Fig. 2, the measured vorticity ®eld for the ¯ow between the compressor blades is shown as a grey value image (original in color). In this image clearly areas may be discerned which have a de®nitely dierent vorticity value compared to their surroundings. The term `structure' may be used to describe coherent, organized motions occurring in the ¯ow. An often
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57
Fig. 3. (a) Source image. (b) Reconstructed image (rms error 0.17). Simulated source image and reconstruction of source image from projections at ÿ12, 0, 12, 78, 90 and 1028 using the same Fourier±Bessel expansion as in experiments. (M = 4, K = 24).
used de®nition in this sense was given by Robinson [25], who de®ned a coherent motion or structure as [26] a three-dimensional region of the ¯ow over which at least one fundamental ¯ow variable (velocity component, density, temperature, etc.) exhibits signi®cant correlation with itself or with another ¯ow variable over a range of space and/or time that is signi®cantly larger than the smallest local scales of the ¯ow. Organized structures are found to exist in several turbulent ¯ows and are responsible for turbulence production and transportation. A better understanding of the dynamics of these structures is a major issue for
the improvement of turbulence modeling and the implementation of eective turbulence control. The study of large scale turbulent structures is thus important because they appear to dominate several turbulent ¯ows.
4. Results HIT allows the study of both absolute as well as dierential turbulent density structures [7]. The resolution that may be obtained for determining these structures, depends on the experimental viewing con®guration that is used to obtain the data, as described in the previous section. The resolution of the tomo-
Fig. 4. Sinogram for underexpanded, supersonic case of the 6 absolute phase maps. Viewing order from left to right: 78, 90, 102, ÿ12, 0 and 128, viewing area: 7 14.4 mm2 (101 207 pixels) per view starting immediately below the nozzle exit.
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Fig. 5. Iso-surface at a density of 1.35 kg/m3 for the supersonic case. Volume: 5 5 5 mm3, starting immediately at the nozzle exit, ¯ow direction: top-down.
graphic system has been evaluated theoretically for the speci®c tomography con®gurations used in this experiment using a numerically generated test ®eld. The test ®eld has also been used in previous studies by Watt [17,18,24]. It represents a turbulent ¯ow ®eld, containing both the smooth and ®ne scale features (Fig. 3a). Fig. 3(b) shows the reconstruction of the simulated turbulent ®eld using 6 views, which are unevenly spread in two orthogonal fans, creating projections at ÿ12, 0, 12, 78, 90 and 1028. This reconstruction was made using 4 harmonics in azimuthal direction and 24 in radial direction, approximating the Nyquist criteria for a 6 view con®guration with 101 pixels in each view. As can be seen from this reconstruction, the ®ne scaled features present in the source image are not reproduced. However, the large scale overall features are found even using this low number of views. To test what kind of results may be obtained in actual experiments some test ¯ows were generated using a 450 mm long cylindrical tube, with an internal diameter of 3 mm. At suciently high pressures at the nozzle exit, expansion to supersonic conditions may be expected to result in the well-known diamond-shaped pattern for the density distribution in the resulting free jet, which will decay due to free shear layer inter-
actions. At the supersonic condition studied here the pipe produced an underexpanded jet ¯ow which is stable when leaving the pipe but becomes unsteady at about 2.5 diameters from the nozzle exit as can be seen in Fig. 4 where 6 phase maps obtained from dierent viewing directions, corresponding to those used to generate the simulated result shown in Fig. 3(b), are shown in the form of a so-called sinogram. These six absolute phase maps were made at the same instant of time, thus providing information on the instantaneous behavior of the ¯ow ®eld. The individual phase images were unwrapped, corrected for tilt and magni®cation to obtain the projection measurements at 101 points in each of a number of planes centered in the nozzle. These corrected projections were then collected into the sinogram, which contains the projections obtained from a single volume. Each plane in the sinogram was then reconstructed using the Fourier± Bessel technique described in Section 3. Each image in the sinogram image of Fig. 4 is actually produced from two 512 512 images grabbed by the CCD camera from the same hologram in order to improve the resolution. Thus, smaller features may be resolved while retaining a large ®eld of view, in this case especially in the direction of the ¯ow.
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Fig. 6. (a) Iso-surface at a density of 1.47 kg/m3 (also shown are markers that were put in after the reconstruction). (b) Slice through y, z-centerplane of reconstruction. Tomographic reconstruction of free jet ¯ow using 6 view set-up. Volume: 5 5 5 mm3, starting immediately at nozzle exit; ¯ow direction: top-down.
The phase values are rendered in grey tones, where the (zero-phase-dierence) background is rendered in a light shade of grey and the phase dierence increases from grey through white to black. The maximum phase dierence in each view is about 1.26 2p, corresponding to a maximum average density inside the jet of 1.29 kg/m3 above ambient (with l=693.4 nm and based on a ¯ow width of 3 mm). In the phase maps of Fig. 4, going from the nozzle along the centerline, ®rst a decreasing phase can be seen agreeing with an expansion in the nozzle exit. Further downstream the phase increases again. After re¯ection of the shock waves at the free jet boundary, subsequent expansion causes the (integrated) density to decrease again. This repeated compression±expansion sequence results in the familiar diamond shaped pattern for the integrated density in the steady part of the ¯ow ®eld. After some streamwise distance the ¯ow develops an instability, which is re¯ected by the wavy pattern in the lower part of the phase map of Fig. 4. Fig. 5 shows an iso-density-surface of the 3D reconstructions for the upper part of this volume at a density level of 0.15 kg/m3 with respect to ambient. For this reconstruction 101 101 pixels were used per viewing direction, representing a volume of 5 5 5 mm3. In Fig. 5, from the shape of the iso-density-surface, two compression areas can be recognized, one starting immediately at the nozzle exit, the other somewhat further downstream (cf. Fig. 4). Also an increasing complexity of the ¯ow can be seen.
As a further example of the results that may be obtained in actual experiments, in Fig. 6 a tomographic reconstruction is shown of a typical subsonic free jet using the viewing con®guration of Fig. 3(b) and the same ¯ow generator as used for the previous results. Again all six absolute phase maps used in the reconstruction were recorded at the same instant on the holographic plate. Prior to reconstruction, the projections in the sinogram were ®ltered with a twodimensional lowpass ®lter. For the reconstruction shown here, 101 101 pixels were used per view, which in this case represents an area of 5 5 mm2 starting immediately below the nozzle exit. Fig. 6(a) shows an iso-density surface, while Fig. 6(b) shows the center slice through the ¯ow ®eld. The iso-density surface shows that the jet-shape evolves from a basically circular form close to the pipe exit, into a noncircular shape starting about one pipe-diameter from the exit. This spatially non-uniform character of the ¯ow is seen more clearly from the center slice through the jet shown in Fig. 6(b). Fig. 6(b) shows the density values in the form of grey scales. Thus, several light shaded connected areas can be seen, representing dierent density structures that are present in the ¯ow. From Fig.6(b) an increase in structure size and strength can be seen with increasing distance from the pipe exit, indicating an increasing turbulence level. A series of these slices, centered around the volume axis in the x, z-plane is shown in Fig. 7. These slices re¯ect the 3D character of the recorded ¯ow ®eld. Following the structures in Fig. 7 from (a) to (e) it can
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Fig. 7. (a) Slice at ÿ0.7 mm from center of nozzle; (b) slice at ÿ0.2 mm from center of nozzle; (c) slice at +0.2 mm from center of nozzle; (d) slice at +0.7 mm from center of nozzle and (e) slice at +1.2 mm from center of nozzle. Slices around the center of studied volume, taken in the x, z-plane (originals in color).
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Fig. 8. (a) Source image. (b) Reconstructed image (rms error: 0.09). Simulated source image and reconstruction of source image from projections at ÿ70.2, ÿ47.1, ÿ31.3, ÿ16.9, 0, 16.9, 30.8, 47.1 and 69.78, using the same Fourier±Bessel expansion as in experiments (M = 9, K = 23).
be seen that in (a) the structures are mostly present on the top left side, near the nozzle exit, while at position (e) structures are seen on the lower right side furthest from the nozzle. Furthermore, it becomes clear that the actual ¯ow is not centered around the axis of the nozzle in this plane, but appears to be shifted by
about 0.2 mm, which also appears to be the case for the supersonic ¯ow discussed earlier. This 6 view con®guration is obviously not ideal, as may be judged directly from the simulation results shown in Fig. 3. The number of views is limited and also the angular spread is limited, resulting in sparse
Fig. 9. (a) Iso-density surface showing columns connecting compression regions, density level: 0.32 kg/m3 with respect to ambient. (b) Axial slice through edge of jet; x, z-plane. Tomographic reconstruction of underexpanded free jet ¯ow, based on 9 views. ps=3 bar, pipe inner diameter d = 6.5 mm, volume: 20.4 20.4 20.5 mm3. Flow direction: top-down.
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Fig. 10. Tomographic results for oscillating free jet ¯ow showing iso-density-dierence surfaces. Bright regions: density dierence of 0.14 kg/m3, dark regions: ÿ0.14 kg/m3. D T =76 ms, Flow direction: top to bottom, volume 17 17 16.88 mm3.
sampling of the investigated ®elds. Furthermore, this con®guration also provides an uneven sampling, reducing the obtainable accuracy even further. To improve the quality of the reconstructions that may be obtained, a tomographic system has been developed, which records 9 simultaneous views, again at the same instant of time [9]. The beams produced in this tomographic set-up simultaneously traverse the test ®eld at angles of ÿ70.2, ÿ47.1, ÿ31.3, ÿ16.9, 0, 16.9, 30.8, 47.1 and 69.708, producing a tomographic system which is symmetrical with respect to the z-axis (direction of original object beam). This con®guration provides views, which are roughly equally distributed over a range of about 140 degrees. The ®eld of view of the present set-up is about 40 40 40 mm3. Fig. 8 shows the results that may be obtained using the Fourier±Bessel technique for the simulated turbulent ®eld for this 9-view con®guration. In this case 9 harmonics were used in azimuthal direction and 23 in the radial direction. The resolution that may be obtained using this con®guration is clearly greatly improved, as now also smaller scale structures appear to be resolved. From Fig. 8 it can be seen that the size of the structures that can be discerned using this setup is about 1/50 of the original projection size used for the reconstruction, when the object covers about 40%
of the projection size. In that case, therefore, the size of the structures that can be resolved potentially is about 1/20th of the size of the original ®eld. In the cases shown here this corresponds to about 250 mm. Fig. 9 shows a tomographic reconstruction of an underexpanded free jet ¯ow. This ¯ow is generated by a con®guration consisting of a high pressure line which supplies a settling chamber with an inner diameter of 35 mm, which contracts smoothly into a round tube which is 60 mm long. Thus, the studied jet ¯ow issues from a straight pipe with a 6.5 mm inner diameter. The tomographic reconstruction shows the ¯ow to be essentially circular in the diamond shaped compression regions, whereas the areas connecting the compression regions show some clearly three-dimensional features (Fig. 9a). These ¯ow features are as would be expected for a ¯ow from a (relatively long) round pipe. Because of the pipe length, a turbulent ¯ow will be generated, which may contain irregular patterns even close to the exit, especially in the shear layer regions. Fig. 9(b) shows an axial slice through the outer layer of the jet at about 3.2 mm from the center plane. This image clearly indicates the unsteady, oscillatory character of the ¯ow ®eld, which increases in strength as it progresses downstream. A comparison of the tomographic results obtained for this ¯ow to
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results obtained for the same ¯ow using an experimental Rayleigh scattering technique and a numerical Euler technique show an encouraging similarity [27]. As the Euler technique only produces steady ¯ow results and the Rayleigh results were averaged over several laser shots, some of the dierences that were found may be due to the actual unsteady character of the ¯ow, which is only captured by the tomographic technique. An indication of this unsteady behavior was also found by Dam et al. [28], in their Rayleigh scattering measurements when comparing images obtained from single laser pulses. The density areas that may be identi®ed in the tomographic results such as shown in Fig. 7 may possibly be interpreted as coherent density structures (cf. Fig. 2). To positively identify these as coherent structures however, their behavior in time should also be followed. By recording dierential interferograms using the rapid-switching facility of the DHI set-up [7], it is possible to measure the density changes that occur in the ¯ow over an adjustable time-interval. When coherent density structures are imagined to be density volumes travelling through the ¯ow [29], the resulting dierential density results may be interpreted as follows: When a (single) structure moves in an otherwise homogeneous area (idealized example, for time intervals which are long enough for the structure to have moved a distance greater than its own length), the phase map consists of two structure images, displaced in the ¯ow direction. One has a decreased phase and the other an increased phase, more or less similar in form, assuming that the structures are only displaced and not deformed in the time interval. As an illustration of the capabilities of the combination of the 9-view tomographic method and the quantitative double-pulsed interferometry, Fig. 10 shows a tomographic reconstruction of a dierential measurement of a free jet ¯ow that is oscillating at a frequency of 6550 Hz [30]. In Fig. 10, a dierential recording at a pulse-interval of half the oscillation period is shown. The dierential recording shows volumes with increased and decreased density. This may be interpreted as volumes between which the structures have moved. In the case shown here, it is seen that structures are displaced vertically as well as in-plane. Thus, it is shown that the tomographic doublepulsed technique may be used in order to localize density structures and to determine and investigate their shape, strength and velocities (at least for the larger structures). In the case shown here, the velocity with which the structures have moved may be estimated roughly to be in the order of 70 m/s.
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5. Discussion The actual resolution that has been obtained using the 9 view tomography system is about 1 mm in size while the density level that may be discerned is about 0.04 kg/m3 [27]. The resolution of this system is limited by the number and spread of the views and also by the number of projections per view. Generally, it is found that the improvement in reconstruction accuracy decreases with increasing these parameters [31,32]. Therefore, increasing these beyond a certain limit barely improves the reconstruction as the additional projections are not suciently independent from the existing ones to provide non-redundant information. For a simulated turbulent ®eld, containing no shocks nor opaque objects, Watt and Conery [24] found that no signi®cant improvement of the reconstruction will be obtained by using more than 12 views. Holographic data has been used in this paper to tomographically reconstruct and visualize the three dimensional turbulent structure within a transonic jet. The structure is of interest because of the detail which has been obtained. Having used holographic interferometry and tomography to de®ne the three-dimensional instantaneous turbulent structure within the transonic ¯ow, it is of interest to explore both its spatial and temporal behavior. The tomographic system described in this paper was designed for application, with minimum alteration, to the high speed wind tunnels at Delft. To achieve this the system needed to be both compact and rigid. With the advent of high resolution high speed digital cameras it is now possible to replace the holographic recording medium. In this way a real-time interferometric system may be created where for each viewing direction the phase-stepped phase maps are simultaneously recorded. A further simpli®cation is possible by the use of a shearing interferometer [33]. Such systems are far less sensitive to vibration or mechanical path changes and are optically very simple to build. Also, although the measured projections may be more noisy in the case of shearography than for interferometry, recently Watt [18] showed, based on numerical simulations, that gradient type reconstructions are no more susceptible to measurement noise than projection techniques. The holographic interferometry (HI) described in this paper is not real time. However, by using either of the approaches described by Pellicia and Watt [34] or Fomin et al. [35], combined with the knowledge of the structure obtained from HI, it is possible to see how a real time system could be developed. Pellicia and Watt replace HI with shearography, which uses Fourier shift theory to unwrap the phase maps, prior to its tomographic reconstruction. This approach is an order of magnitude less sensitive than phase stepping but does
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allow a high speed capture approach to the problem and also a fully automated data extraction. Shearography also provides a dierential measurement, which requires some interpretation with respect to the turbulent structures. Its main diculty however, is the experimental complexity of the system, which needs to extract multiple views. This restricts its application to open ¯ows, which do not have a viewing limitation. An alternative, statistical, approach is proposed by Fomin et al. [35]. They use the refractive index ray bending which occurs as the light passes through a turbulent structure to de®ne its position in the direction of light path propagation. To achieve this an auxiliary measurement is used, which is based on the average size distribution of the turbulent structure. This is then used to model the ¯ow structure and then match the positional bending of the ray by calculation. The refractive index bending is a source of error in the HI case and is ignored as being much smaller than the ray path phase change through the structure. However, in the current case HI has been used to de®ne the size and shape of the turbulent structures in three dimensions. In particular it has shown the amount of structure within the jet is small and relatively well de®ned. Thus, by using this data, a model of shapes and structures could be used to increase the accuracy of the statistical method. The advantage being that once validated in this manner, the experiment only requires one direct view through the ¯ow ®eld. This opens up the possibility of extracting both the temporal and three dimensional spatial evolution of complex turbulent high speed structures in a greater detail than previously obtained. 6. Conclusions The results presented here indicate that the tomographic interferometry technique is able to provide quantitative information regarding the three-dimensional structure of real, unsteady ¯ows. Large-scale, unsteady density structures may be identi®ed, at density-dierences down to 0.04 kg/m3, showing the complete three-dimensional temporal behavior of ¯ows. Details of the spatial structural evolution of transonic free jet ¯ows have been obtained. The results show an increase in the size and complexity of this structure with distance along the jet. The result also presents a visualization of the oscillatory unsteady ¯ow nature in the outer layers of the jet. By changing the pulse-interval in double-pulse experiments, unsteady or oscillating ¯ows may be followed in time. This gives insight into the behavior of unsteady ¯ows. A single instantaneous image by itself does not provide information on the temporal be-
havior, whereas dierential measurements only show the changes in the ¯ow. Combined, however, they oer a understanding of both the spatial and temporal ¯ow behavior. Furthermore, from dierential measurements, coherent structures may not only be identi®ed, but also their velocity can be determined. Thus, the speed of mixing and transportation phenomena may be studied. Also using this technique, it is possible to extract turbulence time-scale information from dierential interferometric results for an arbitrary compressible ¯ow ®eld. Most importantly, the tomographic technique has provided an instantaneous three-dimensional visualization of both the steady and unsteady behavior of the ¯ow which could not have been achieved using more conventional techniques.
Acknowledgements The technical support of Frits Donker-Duyvis and Eric de Keizer of the Laboratory for High Speed Aerodynamics of the TU Delft is greatly appreciated. DWW's contributions to this project were supported by NSF Grant Number ECS8910350 and the NATO Scienti®c Research Programme. Part of BHT's work was funded by the Dutch Organisation for Scienti®c Research (NWO) through Projectnumber DLR33.3109 of the Technology Foundation (STW).
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