Tomographic reconstruction of temperature fields acquired with the laser speckle BOS technique

Transcription

1 Tomographic reconstruction of temperature fields acquired with the laser speckle BOS technique P. Bühlmann 1*, J. Schenker 1, T. Rösgen 1 1: Institute of Fluid Dynamics, ETH Zurich, Switzerland * Correspondent author: buehlmann@ifd.mavt.ethz.ch Keywords: Background Oriented Schlieren, Speckle Imaging, Wall-bounded flows, Tomographic reconstruction ABSTRACT The laser speckle based background oriented schlieren technique (speckle BOS) is used to study spatial temperature fields of a circular and an elliptic jet. A specific advantage of the speckle BOS method is the possibility to align the laser beam with the camera (double-pass BOS imaging) resulting in an increase of sensitivity. Details of the double pass mode are discussed and the increase of sensitivity is quantified for divergent illumination. To evaluate the performance of the speckle BOS technique in a tomographic imaging environment, a round, turbulent heated jet was simultaneously recorded in the double pass and the (standard) single pass modes and tomographically reconstructed with an iterative MART algorithm. As a more complex geometry, the temperature distribution of the flow above a rectangular nozzle located close to the speckle screen is finally analyzed using double pass measurements. 1. Introduction The background oriented Schlieren technique is a well-established method to generate quantitative data of refractive index variations in optically transparent fluids. Meier and Roesgen (2013) extended this method by introducing laser speckle illumination as a virtual background image (speckle BOS). In contrast to the classical BOS technique, the BOS reference pattern is replaced by a rough screen reflecting coherent light, which finally interferes on the imaging sensor s surface forming a speckle pattern. This pattern exhibits a random intensity distribution, which can be characterized by statistical means (Goodman 2007) and whose displacement can be tracked with the usual PIV algorithms. The interference properties of the speckle pattern lead to several improvements compared to the classical BOS technique. - The camera can be focused at any position independent of the reference screen position. Thus, the reference screen can be placed closed to the measurement object while

2 maintaining sensitivity. Furthermore, the overall sensitivity can be adjusted independent of the imaging geometry. - In the classical BOS technique, the reference pattern is illuminated with incoherent, diffuse lighting. Speckle BOS imaging allows for the alignment of the laser beam with the camera (double-pass mode) resulting in an increase of sensitivity. The speckle BOS method is currently being implemented for tomographic reconstruction of the temperature distribution in the backlayering zone of a simulated tunnel fire. This flow regime is located close to the tunnel ceiling, where a proper application of the standard BOS method is difficult. Exploiting the favorable optical properties of the speckle BOS method, an approach to reconstruct spatial and temporal temperature distributions of near wall flows is being developed. Since illumination from the side is not possible in this scenario, double pass BOS imaging is used. In the following section, the double pass mode will be discussed and, in analogy to the classical BOS technique, a relation between the measured speckle pattern shift and the refractive index gradient using divergent illumination is introduced. To evaluate the performance of the speckle BOS technique in a tomographic imaging environment, a round turbulent heated jet was simultaneously recorded in the double pass and the (standard) single pass modes. Temperature distributions inside the jet are tomographically reconstructed using a MART algorithm ( multiplicative algebraic reconstruction technique ). The corresponding results are presented in section 3. As a more complex test case closer to the final backlayering geometry, the temperature distribution of the flow above a rectangular nozzle located close to the speckle screen is analyzed as well using the validated double pass arrangement. 2. Double-pass BOS imaging A typical double-pass BOS imaging set-up is illustrated in Fig. 1. It can be divided into a first pass (distorted illumination only) and second pass contribution (distorted reflections only), as displayed in Fig. 2. The optical set-up of the second pass mode (Fig. 3) corresponds to the standard BOS technique whereas the geometry of the first pass mode corresponds to the measurement configuration of classical speckle photography (e.g. Walsh and Kihm 1995).

3 Fig 1: A typical double-pass BOS imaging set-up. Both incident and reflected beams are deflected by the test section distortions. Fig 2: The double-pass imaging set-up unwrapped into the first pass (speckle photography) and second pass mode (standard BOS). Fig 3: A speckle BOS recording set-up (second pass mode). In contrast to the standard method using a reference pattern, the speckle screen can be placed out of focus.

4 The relation between the pattern shift δs pattern and the deflection angle ε for the well-known standard BOS technique (second pass mode) is reported in detail by Goldhahn and Seume (2007). Assuming small deviation angles from the centerline, this relation can be approximated by δ " = Ml tan ε, M =, -./0,, (1) where M describes the magnification of the imaging lens, f its focal length and m the distance between lens and test section. The parameter l represents the distance between test section and the position where the imaging system is focused. For focusing at infinity, Eq. 1 approaches an asymptotic value. lim δ " = f tan ε (2) - 5 For speckle photography (first pass mode), the displacement δf and the deflection angle ε are linked by δ, = M l s tan ε, (3) where s describes the distance between test section and speckle screen (Fomin 1998). The pattern shift of the overall double pass system is given by the sum of the first and second pass contributions δ = δ, + δ " = M 2l s tan ε, lim δ = 2fε (4) - 5 which corresponds to the relation introduced by Meier and Roesgen (2013). So far, we considered parallel illumination. For larger measurement volumes, the laser beam has to be expanded resulting in a divergent beam pattern. Following the argumentation of Debrus et al. (1972), the formalism can be extended for this case. The undistorted reference beam impinges on the speckle screen with a local angle β leading to a modified relation between δ and ε (Fig. 4).

5 Fig 4: Geometry of the incident light beam deflection for divergent illumination. δ, = M l s tan(θ) (5) Using trigonometric relations, β can be approximately related to ε close to the centerline in terms of the optical parameters d (distance between light source and test section) and s. θ ε β 1 " A." ε (6) For small deflection angles (tan (ε) ε), the double pass relation between δ and ε approximately yields δ = l + B C0D B.D Mε, lim δ = A A." fε. (7) A comparison of Eq. 1, 2, 4 and 7 reveals the increased sensitivity of the double pass mode compared to the single pass mode. For regions close to the wall (s 0), Eq. 7 simplifies to Eq. 4, which describes the double pass sensitivity for parallel illumination. 3. Tomographic reconstruction of temperature fields Considering the integrating nature of the schlieren technique, a tomographic analysis of the measurement volume is necessary to reconstruct spatial refractive index or temperature distributions. As a first step, a round, turbulent heated jet is simultaneously recorded in the double pass and the (standard) single pass modes. Depending on the measurement geometry, the relations 1, 2 or 7 as well as reference values for the temperature T0 and the refractive index n0 of ambient air were used to obtain quantitative data.

6 3.1 BOS diagnostic set-up The geometry of the experimental set-up is illustrated on Fig. 5. A diode-pumped solid state laser (Coherent Verdi, λ=532nm, Pmax=5W) generates a coherent, vertically polarized light beam, which is expanded by a 10x telescope followed by a plano-concave lens (f=50mm). The beam is redirected by a polarizing beam splitter through the heated round jet (first pass). A varnished wooden or Teflon plate serve as speckle screen, where the incident light is scattered and depolarized. After being deflected by the heated jet (second pass), the horizontally polarized component of the reflected beam passes the beam splitter and is finally recorded by a monochrome digital camera (IDS ueye 2230SE, 1024x768) with a 50mm objective lens focusing at infinity. As reference measurement, an identical camera is placed at the same distance from the jet at a different angle simultaneously recording pattern distortions in the well-known single pass mode. In order to avoid peak-locking effects, the f-number of the imaging lens is set to 16, increasing the effective size of the individual speckles. Fig 5: Speckle BOS diagnostic set-up illustrating the simultaneous single and double pass measurements. 3.2 Tomographic approach The BOS deflection angle ε and hence the recorded pattern shift δ are linked to the integrated refractive index gradient along the line of sight by δ(x, z) I JK L N,O n x, y, z dy, (8)

7 where the constant K is given by Eq. 2 or 7 depending on the measurement geometry. By taking the divergence, a Poisson equation for the integrated refractive index n Int is obtained n TKU x, z n x, y, z dy = Kn X δ x, z, (9) which is solved by an in-house MATLAB algorithm. The pattern shifts at the boundary and the index of ambient air n 0 in undistorted edge regions serve as boundary conditions. Assuming constant pressure, the ideal gas law and the Gladstone-Dale relation link temperature and refractive index distributions. Z = K L0I Z L K0I (10) Three-dimensional refractive index and temperature information is extracted from n Int(x,y) using tomographic reconstruction, where the test section is interrogated from different directions. Various methods for tomographic BOS analysis have been demonstrated (Raffel 2015). Due to their robustness to noise, we focus on algebraic reconstruction methods. In this approach, the tomographic problem is described by a set of linear equations, where usually the number of equations and unknowns significantly differ. A description of the algebraic approach is given for instance by Herman and Lent (1976) or Elsinga et al. (2006). To solve the linear system, iterative solution algorithms based on different optimization criteria are used. So far, the most accurate results in this work were obtained by a Lent2 MART algorithm (Verhoeven 1993). Exploiting the symmetry of the round jet, Abel inversion could also be used. However, this approach is considered susceptible to noise (Venkatakrishnan and Meier 2004, Smith and Keefer 1988), which also became apparent during the tomographic analysis of the round jet. 3.3 Temperature fields of a circularly symmetric jet Due the rotational symmetry of the jet, image acquisition from one projection angle is sufficient. Overall, 500 flow images are recorded with an image acquisition rate at 5Hz. The apparent pattern shifts are extracted by an in-house iterative image deformation PIV algorithm (Scarano 2002). An instantaneous displacement field is illustrated in Fig. 6. After time averaging of the displacement field, the rotation axis is detected and its off-center shifts are corrected in the frequency domain following the approach of Smith and Keefer (1988). Furthermore, a low pass filter is applied removing high frequency noise while preserving the lower frequency signal information. The preprocessed pattern shifts are replicated for the MATLAB MART algorithm,

8 18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics LISBON PORTUGAL JULY 4 7, 2016 which is based on the Toolbox Tools for 2-D Tomographic reconstruction written by Ligong Han. Fig 6: Typical speckle pattern with a superimposed instantaneous displacement field of the circular jet. The mean temperature profiles along the centerline of the circular jet for single and double pass imaging are compared in Fig. 7. Thermocouple measurements serve as reference data. Figure 8 illustrates the reconstructed radial distributions of mean temperature for different heights above the nozzle exit. Fig 7: Mean temperature profile of the circular jet along the centerline (r=0) for single and double pass recordings. Fig 8: Reconstructed radial distributions of the circular jet for different heights above the nozzle exit.

9 3.4 Temperature fields of an elliptic jet As a more complex geometry, resembling the final backlayering flow scenario, the temperature distribution in the flow above an elliptic nozzle (aspect ratio 7:1) located close to the speckle screen was analyzed using double pass measurements. The nozzle was successively rotated, and images sequences (500 frames each) for 9 different projections at angles between 20 and 180 were recorded. Exploiting the symmetry of the elliptic jet, a complete data set with 18 different projections between 20 and 360 is obtained. The shape of the jet is visualized in Fig. 9 by mean temperature isosurfaces for 56 C, 50 C and 40 C. The lateral spreading of the jet for increasing height is clearly visible. The mean temperature profile in Fig. 10 of the jet along the centerline agrees with thermocouple data. Fig. 11 and Fig. 12 illustrate reconstructed distributions of the mean temperature along the long (x=0) and the short (y=0) axes for different heights above the nozzle exit. The inconsistencies visible at the edges and at the centerline of the jet may be due to measurement noise or tomographic reconstruction artifacts and will be investigated further. Fig 9: Mean temperature isosurfaces of the elliptic jet for 56 C (blue), 50 C (red) and 40 C (green). Fig 10: Mean temperature profile of the elliptic jet along the centerline (x=0, y=0).

10 Fig 11: Reconstructed distributions of mean temperature of the elliptic jet along the short axis (y=0) for different heights above the nozzle exit. Fig 12: Reconstructed distributions of mean temperature of the elliptic jet along the long axis (x=0) for different heights above the nozzle exit. 4. Conclusions The speckle BOS technique was successfully employed to measure spatial temperature fields of a heated circular and a heated elliptic jet. A mathematical model for double pass imaging with a divergent light source was introduced and successfully employed in the tomographic reconstruction of the spatial temperature distributions. The reconstructed mean temperature profiles agree with independent thermocouple measurements. Further development of the Speckle BOS technique will focus on minimizing the number of projections required for nearly two-dimensional flows as occurring in tunnel fire backlayering zones. This will facilitate the use of multiple cameras for simultaneous and time-resolved recording of the temperature fields. References o Debrus S, Françon M, Grover CP, May M, Roblin M (1972) Ground glass differential interferometer. Appl Opt 11(4): o Elsinga GE, Scarano F, Wieneke B, von Oudheusden B W (2006) Tomographic particle image velocimetry. Exp Fluids 41: o Fomin NA (1998) Speckle photography for fluid mechanics measurements. Springer, Berlin

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