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2D index field measurement with a pyramidal sensor
Béatrice Sorrente, Bruno Fleury, Jean-Marc Conan
To cite this version:
Béatrice Sorrente, Bruno Fleury, Jean-Marc Conan. 2D index field measurement with a pyramidal sensor. ISFV 18, Jun 2018, ZURICH, Switzerland. �10.3929/ethz-b-000279203�. �hal-02176399�
Research Collection
Conference Paper
2D index field measurement with a pyramidal sensor
Author(s):
Sorrente, Beatrice; Fleury, B.; Conan, J.M.
Publication Date:
2018-10-05 Permanent Link:
https://doi.org/10.3929/ethz-b-000279203
Rights / License:
In Copyright - Non-Commercial Use Permitted
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18th International Symposium on Flow Visualization Zurich, Switzerland, June 26-29, 2018
ISFV18 – Zurich, Switzerland – 2018 1
2D INDEX FIELD MEASUREMENT WITH A PYRAMIDAL SENSOR B. Sorrente1,c, B. Fleury1, J.M. Conan1
1ONERA, Châtillon, 92320, France
cCorresponding author: Tel.: +33146734853; Email: [email protected] KEYWORDS:
Main subjects: flow visualization Fluid: subsonic flows
Visualization method(s): pyramidal wavefront sensor Other keywords: wavefront sensor, aero-optic effects
ABSTRACT: This paper presents a four channel schlieren sensor called pyramidal wavefront sensor (PWS) able to estimate the flow field index from phase gradient data. The resolution of the instrument is directly given by the pupil sampling path. The new sensor has been implemented on the SONIQUE wind tunnel for preliminary experimental tests. The demonstration has been realized by analyzing the wake flow around a circular cylinder. A very good agreement is obtained with the phase measured with digital holography method, thus validating the data integration process and showing that PWS is an interesting alternative to
optical methods dedicated to flow characterization.
1 Introduction
Computational fluid dynamics codes are time consuming and very costly. They still need to be validated by optical methods, especially in presence of unsteady wake, in high Reynolds number cases or for atypical configurations. Physical parameters of the fluid flow can thus be experimentally estimated at small scales and low exposure time. Digital holography [1], schlieren and back oriented schlieren (BOS) [2], which is indeed a wavefront sensor, are among the principal methods adopted by the aerodynamicists. From optical measurements taken at different points of view, it is possible to reconstruct 3D index or gas density of the whole analyzed volume.
ONERA has recently completed this range of technics by developing a pyramidal wavefront sensor (PWS) devoted to flow characterization. The pyramidal sensor described in this paper is a Foucault like wavefront sensor but it could also be called a four channel schlieren sensor. It was originally proposed for the first time by R. Raggazzoni for astronomy applications [3]. Generally schlieren methods have provided qualitative informations of the flow field index. But PWS can provide absolute value of the refractive index since two phase gradients are delivered instead of one phase gradient for the classical schlieren technics. To achieve this goal ONERA has developed a reconstruction method that efficiently allows to estimate the phase map. The configuration of the PWS is described in detail in the first part of the paper. Then we report experimental measurements.
2 Description of the Pyramidal Wavefront Sensor
A schematic view of the PWS is shown on Fig. 1. A Cobolt pulsed laser emitting at 638 nm coupled to a 400 m multimode fiber with a 100 µm core diameter generates a spatially incoherent source installed at the focal plane of the first lens. In the original concept a point source is placed at the object plane and a sweeping mirror is temporally modulated to elongate the size of the image on the edge of the pyramid
B. Sorrente, B. Fleury, J.-M. Conan
allowing thus to adjust the gain of the error signal [4]. Due to the very small exposure times required to study aero-optics phenomena and the high amplitudes of these effects, we investigate here a modified version of the PWS where the gain is adjusted by optimizing the diameter of an extended incoherent source and the focal lens placed in front of the pyramid.
A collimated beam passes then through the fluid flow where an entrance mother pupil is located. This beam is then focalized on the edge of a four facets pyramidal prism that splits the focused beam in four parts. A relay lens placed behind the prism produces on a camera four images of the pupil plan located inside the flow. The four images will be called here Iij. An ecartometric error signal is then computed for each common pixel of the four daughter pupils:
𝑆𝑥𝑗 = (𝐼1𝑗+𝐼3𝑗)−(𝐼2𝑗+𝐼4𝑗)
𝐼1𝑗+𝐼2𝑗+𝐼3𝑗+𝐼4𝑗 (1)
𝑆𝑦𝑗 = (𝐼1𝑗+𝐼2𝑗)−(𝐼3𝑗+𝐼4𝑗)
𝐼1𝑗+𝐼2𝑗+𝐼3𝑗+𝐼4𝑗 (2)
Fig. 1. Schematic view of the pyramidal wavefront sensor.
The term Iij is the intensity relative to the jth pixel in the ith sub-pupil (1 ≤ i ≤ 4). In the geometrical approximation it can be shown that the distribution of the illumination in the four pupil images depends on the aberrations of the wavefront and is proportional to the first derivatives of the wavefront [5][6]:
𝑆𝑥 = 𝜆
2𝜋𝜃(𝜕𝜑(𝑥,𝑦)
𝜕𝑥 ) (3)
𝑆𝑦 = 𝜆
2𝜋𝜃(𝜕𝜑(𝑥,𝑦)
𝜕𝑦 ) (4)
and are respectively the wavelength and the angular radius of the extended source while represents the distorted wavefront that has to be analyzed. In absence of turbulence, a reference error signal is recorded and then substracted to the signals obtained in presence of the flow field.
By construction, the PWS signals are built to be comprised between -1 and 1 and saturates in presence of high amplitude aberrations as illustrated on Fig. 2. So expressions are only valid in the linear regime of the PWS. The linear domain is fairly large, but for high amplitude aberrations the PWS may underestimate the distortion of the phase.
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ISFV18 – Zurich, Switzerland – 2018 3
Fig. 2. Illustration of the influence of the object size on the linearity domain. Front view of the pyramid for two object sizes (left) - Typical pyramid response for the two object sizes in presence of a tilt aberration (right).
So the dynamic and the sensitivity of the sensor must be optimized by adjusting the angular diameter of the extended object and the specification of the lens placed in front of the pyramid.
Finally the phase is estimated from its two gradient maps using a zonal reconstruction method that we specifically develop for the PWS [8].
Fig. 3. PWS implementation at the SONIQUE wind tunnel.
The PWS has been implemented in the SONIQUE wind tunnel at ONERA (Lille) to study the subsonic near wake unsteady flow downstream a 20 mm circular cylinder at two Mach conditions, 0,55 and 0,8 (Fig. 3) [9]. To optimize the PWS optical parameters, we exploited phase maps obtained by digital holography. This a priori knowledge of the phase range helped us to save valuable time to choose the optical parameters. A phase cartography obtained at Mach = 0,45 with this interferometric method is illustrated on Fig. 4 and it was planned to analyze the flow field inside a 50 mm diameter pupil defined by the red circle.
The best compromise in terms of signal-to-noise ratio for a 300 ns exposure time was to sample each sub-pupil with 128 pixels. The gain and sensitivity of the PWS has been optimized by choosing:
B. Sorrente, B. Fleury, J.-M. Conan
= 40𝜆
𝐷 (5)
where D is the pupil diameter (Fig. 3). In practice the phase has been estimated on a 47 mm diameter sampled with 114 pixels for each sub-pupil leading to a spatial resolution of 412 µm. Intensities have been recorded using a PCO-Edge camera and a full frame frequency of 50 Hz. The power of the laser source was limited to 1 mW in order to minimize the contrast of the speckles under a threshold value of 5%. Table 1 presents the experimental parameters at Mach=0,55 and Mach=0,8.
Fig. 4. Phase map reconstructed by digital holography at Mach = 0,45. The PWS analyses the phase distortion inside the red circle.
Mach 0,55 0,8
Pupil diameter (mm) 47 47
Pupil diameter (pixel) 114 114
Focal lens f0 (mm) 200 150
Exposure time (ns) 300 100
Average ADU number 3000 900
Table 1. Experimental parameters at Mach =0,55 and Mach = 0,8.
A calibration procedure is performed prior to recording data in presence of turbulence for fine phase reconstruction. It consists in introducing calibrated aberrations (tilt or defocus) by moving an optical element and measuring the corresponding pyramidal error signal. Since the phase cartography is z- invariant, the refractive index difference is linked to the phase through the following relation:
∆𝑛 = ∆𝜑𝜆2𝜋𝐿 (6)
L is the wind tunnel width (L = 42 mm). From the Gladstone-Dale relation the gas density profile can then be computed.
D
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ISFV18 – Zurich, Switzerland – 2018 5
3 Experimental results
Fig. 5 displays instantaneous row data of the intensities measured on each sub-pupil for both Mach conditions. The spatial structure of the field gets more complex as the Mach number increases. Blurring effects are observed and shorter exposure time should have been used, but this would reduce too drastically the photometric conditions of the experiment (Table 1).
I1 I2
I3 I4
I1 I2
I3 I4
Fig. 5 – Instantaneous intensity maps obtained at Mach = 0,55 (top) and Mach = 0,8 (bottom).
Fig. 6 exhibits the cartography of the pyramidal signals showing as expected disparate behaviors for each axis. The associated histograms of the pyramidal error signal are represented on Fig. 7. At Mach=0,8 shock waves can be distinguished, but the spatial resolution of the PWS does not seem to be small enough to sample this local phenomena with a sufficient number of pixels. A significant number of Sx and Sy values are closed to 0,9 at Mach = 0,55. This means that the reconstructed phases will be partially and locally underestimated. At Mach = 0,8, the pyramidal error signal spans in a shorter range thanks to the focal lens change.
B. Sorrente, B. Fleury, J.-M. Conan
Fig. 6. Instantaneous error signals of the pyramidal sensor.
Mach=0,55
Mach = 0,8
Sx Sy
Fig. 7. Histograms of the pyramidal error signals.
The instantaneous and averaged reconstructed phases are shown on Fig. 8. At Mach=0,55 a good agreement is demonstrated with the result obtained by digital holography at Mach=0,45 [10]. We checked that the phases obtained with these two optical methods are quantitatively very closed [11]. At Mach =0,8 the exposure time was not small enough to freeze the transonic aero-optic effects. In this regime digital holography is performed with an 8 ns pulsed laser [12]. In order to detect shock waves without blurring effects, future works will be dedicated to the upgrading of PWS in terms of spatial and temporal resolution.
Mach=0,55
Mach = 0,8
Sx Sy
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ISFV18 – Zurich, Switzerland – 2018 7
Mach = 0,55
Mach = 0,8
Fig. 8. Instantaneous and average reconstructed phases.
Conclusion
ONERA has developed a pyramidal wavefront sensor dedicated to flow field analysis that demonstrated very good agreement with digital holography at subsonic speed. Quantitative measurement of the phase allows the estimation of the index field. The PWS represents an easily reconfigurable and robust alternative to other optical methods dedicated to flow characterization. These properties are very important in the vibratory environment of a wind tunnel and when the time available for the tests is limited. To the best of our knowledge, the results presented in the paper show the first application of a PWS to aero-optic domain.
Future works will concern the design of a high spatial resolution PWS for analyzing the flow field around an engine jet observed through a 75 mm pupil diameter sampled with 700 pixels. A specific reconstruction algorithm will be developed for this high level of pixels. Comparisons will be made with results obtained with the BOS technics and digital holography. To increase the temporal resolution of the demonstrator and analyze transonic or supersonic flow, high power diode laser sold with an intrinsic ability to reduce speckles could be supplied.
Acknowledgements
Authors are grateful to Kacem El Hadi from the Laboratoire d’Astrophysique de Marseille (LAM) for the loan of the pyramid, a key component for our tests. Thanks are due to Vincent Michau from ONERA and Jean-François Sauvage from ONERA/LAM for fruitful discussions. The authors are very indebted to Jean-Michel Desse and François Olchewsky from ONERA for their support during the tests.
References
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B. Sorrente, B. Fleury, J.-M. Conan
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