Split-screen single-camera stereoscopic PIV application to a turbulent confined swirling layer with free surface

Particle image velocimetry (Adrian 1991, 2005; Cho 1989) has become the de facto 2D velocity field measurement technique in fluid mechanics. Judicious exploitation of PIV data contamination by out-plane velocity has led to SPIV, which extracts the complete velocity vector on a plane. SPIV can be performed with two optical configurations: lateral and angular displacements (Prasad 2000). In the lateral displacement—or lens translation—technique object, lens, and image planes are parallel and laterally shifted with respect to one another. As a result, the cameras do not need calibration, but the maximum viewing angles are limited. In the angular displacement technique, there is no limitation on the viewing angles, which can be set at their optimum value of 45°, but the cameras need calibration. The object plane can be made in focus without large depth of field (obtained by small aperture lens) if the Scheimpflug condition is implemented. Angular displacement SPIV has been used with (Willert 1997) and without (Lawson and Wu 1997; Westerweel and Nieuwstadt 1991) satisfying this condition.

When feasible, it is beneficial to project both views onto a single camera sensor in SPIV. In particular, if the measurement field has a high aspect ratio, it is simply an efficient use of the camera sensor area. Lateral displacement SPIV technique has been implemented with a single camera (Arroyo and Greated 1991) in which a set of four mirrors are used to project both views onto the camera sensor plane. This configuration is, however, somewhat intricate to implement because it requires the installation of two mirrors between the lens and the sensor chip. Further, it suffers from the limitations associated with lens translation.

Figure 3 shows the angular displacement SPIV configuration employed in this study. Two first surface mirrors and a right angle prism mirror are used to project both views onto a single camera. The viewing angles α₁ and α₂ are nominally 45°. The optical path is equivalent to having two virtual cameras recording the flow. This split-screen imaging technique was first employed for SPIV using a 35-mm film camera by Grant et al. (1991). The film was then digitized for processing. This configuration is inexpensive, easy to install, and align. However, it requires a longer optical path than if using 2 cameras and therefore a longer focal length lens. Scheimpflug condition could be satisfied but it would require the use of 4 prisms with two positioned between the lens and the camera sensor, rendering it rather impractical. This configuration was also employed for stereogrammetry methods (Grant et al. 1995) for flow visualization experiments with reflective flakes (Jacob and Savaş 1995) and for 3D flow measurement (Racca and Dewey 1988; Peskin 1972).

The laser sheet is set to coincide with the tube axis to perform measurements in the radial plane, hence, normal to the mean free surface seen in Fig. 2a. The instantaneous surface shape undulations seen in Fig. 2b, necessitate particular precautions to prevent unavoidable reflections from interfering with measurements, even damaging optical components and the camera sensor.

The approach employed here is to separate the illumination and observation wavelengths by employing fluorescent particles and sharp cutoff filters. The illuminating laser is a double pulsed Nd-Yag laser of 532 nm wavelength (New Wave, Gemini-30). The working fluid is seeded with dry polymer fluorescent particles of 7 μm mean diameter and 1.05 g/cm³ density (Duke Scientific 36-2, proprietary dye). The Stokes shift, or the difference between the maxima of absorption and emission spectra, is 70 nm, i.e. the seeds emit 602 nm red/orange light. A longpass filter with a cutoff wavelength of 570 nm (Schott OG570) is used to isolate the fluoresced light from the laser light.

The cutoff filters are made of a glass matrix embedded with light-absorbing semiconductor colloidal particles, which implies that the absorbed light intensity increases with the filter thickness (Bindra et al. 1999; Oak et al. 1993). To take advantage of this property, two 3-mm thick filters are used in series between the lens and the camera CCD sensor (Fig. 3a). With this optical arrangement, no direct laser light interfered with flow field images (Fig. 4).

The mapping between the image and object planes can be determined by geometrical reconstruction, 2D or 3D calibration (Prasad 2000). Due to spatial constraints imposed by the apparatus, the 2D backprojection calibration method is used here.

The calibration target is a custom manufactured 1.5-mm thick and 25.4-mm square fused silica plate with an index of refraction nearly matching that of the working fluid. On one side of the plate a square grid of 508 μm ± 2.5 μm pitch is laser etched by a laser scribing company—mrlaser.com—(Fig. 5a). The width of the grid lines, essentially shallow channels, are measured with a microscope to be 178 μm. The target is first covered with a ∼150-nm thick layer of gold-palladium by ion sputtering. Then, the plate surface is wiped clean to leave the gold-palladium layer in the channels only, resulting in a target pattern that looks like a dark square mesh on a transparent background (Fig. 5b). To identify corners of the square grid, each view of the calibration target is enhanced by contrast gamma stretching followed by a median filter to remove salt and pepper noise; the views are then cross-correlated with a corner mask, the correlation peaks indicating the corner locations. Subpixel calibration accuracy is obtainable by curve fitting of the local maxima, however this is not implemented here.

Of the two common techniques of 3D displacement construction, mapping of the velocity vectors after SPIV processing or warping the SPIV images before processing (Prasad 2000), the latter was implemented here despite being computationally intensive, since it allows for correcting for misalignments that might result from the calibration step (Willert 1997).

A polynomial mapping function is commonly used for backprojecting the particle images (Prasad 2000). A pair of bilinear equations are employed here where (x, y) is a point on the object plane, and (X, Y) on the image plane (Gonzales and Woods 2001). The coefficients a _i and b _i are estimated through a linear least-mean-square regression analysis using more than 40 points per view of the calibration images. The particles images are then reconstructed using a cubic convolution interpolation technique. Bilinear warping maps nonplanar quadrilaterals (Wolberg 1990). Bilinear mapping preserves straight lines that are vertical and horizontal. Like affine transformations, points along those lines that are equispaced in the source image remain equispaced in the mapped image. Lines that are diagonal, however, are mapped into quadratic curves. Dewarped pictures of the calibration target, with view #1 flipped horizontally, are shown in Fig. 5c. On both views, similar points overlap well within one pixel. Higher-order polynomials were tried, with less satisfactory results. The three displacement components are then reconstructed following Willert’s geometry (Willert 1997).

The flow has simultaneous features of being turbulent, having a solid wall, and possessing a very irregular free surface. Thus, the interrogation of the particles images calls for particular considerations. To accommodate strong gradients that are expected in turbulent flows, special PIV processing algorithms have been developed. They include, but are not limited to, particle image deformation (Huang et al. 1993a, b), super-resolution PIV (Keane et al. 1995), direct estimation of velocity gradients (Mayer 2002), and adaptive Lagrangian parcel tracking (aLPT) by Sholl and Savaş (1997).

aLPT algorithm is employed here. This algorithm utilizes interrogation windows that are advected and deformed according to the local velocity and velocity-gradient fields, improving the quality of the data in regions of strong deformation. The outputs of the algorithm are the two-dimensional velocity vector field and its gradient tensor (which is computed spectrally). The vorticity is determined from the velocity-gradient tensor. The algorithm allows for dynamic resizing of interrogation window. Cross-correlations, derivatives, and filtering are computed with fast Fourier transforms (FFT). For this purpose, the velocity array is centered in the smallest 2 ⁿ × 2 ^m array so that FFT algorithms can be employed for fast calculation. The data are smoothly extended to ‘zero’ values at the edges of the newly constructed FFT domain to insure continuity in the slopes to prevent ringing in the frequency domain. Once in the wave number domain, the transform is multiplied with the appropriate powers of the wave numbers to construct the transforms of the derivatives, which are in turn obtained by subsequent inverse transforms. Derivatives up to 3rd order are obtained for aLPT where they are used to reconstruct deformed interrogation windows. In particular, second order deformations are used where deformation of a rectangular fluid parcel is traced to second order which results in a cushion parcel with curved edges rather than a parallelepiped with straight parallel edges. During these calculations, the raw data are filtered in the frequency domain using a low-pass filter kernel 1 − exp(− 1/k ²) where k is the modulus of the wave vector (normalized with the desired cut off wave number). Before the aLPT processing begins in earnest, a routine PIV correlation technique is used to identify outliers, which are subsequently recalculated at doubled and, if needed, quadrupled the windows. If this process also fails, the outliers are replaced with the mean of their neighboring vectors. The final output of aLPT, however is not scanned for individual outliers, but passed through a general filter as describe above. Because of its accuracy in measuring flow fields in region of strong deformation, aLPT is well suited to work on turbulent flows in general, and hence is appropriate to analyze the data of this experiment.

Before processing the data, particles images are pre-processed to increase the cross-correlation coefficients average value calculated in the PIV analysis software. Since the PIV laser employs two separate laser heads, one frame is usually slightly brighter than the other. The intensity of both frames is then corrected such that their average intensities are equal (Ortega 2001).

In the optical setup of Fig. 3, the actual viewing angles are determined from the calibration images, and are 43.0° and 41.8° for views 1 and 2, respectively. The square mirrors are set at an angle of 54° ± 0.5° from the laser sheet. The nominal magnification is estimated from the calibration images as 0.52.

A double exposure CCD camera (Kodak ES1.0) is employed, which has a resolution of 1008 W × 1018 H pixels and a depth of 8-bits. The PIV image pair acquisition rate is 2 × 15 Hz. A Nikkor 85 mm lens with extension tubes constitute the recording optics. The tandem long-pass filters are positioned inside the extension tubes. The f-stop (aperture setting) is set to f/4 to obtain an acceptable depth of field. The recorded fields of view are about 11.5 W × 17.5 H mm² for both views.

The double-pulsed PIV laser can deliver an energy of 100 mJ per pulse. For this experiment, nearly 95% of this energy is used. The light sheet thickness is about 200 μm. The pulse separation is set to 25 μs for the data presented here. With an out-of-plane average velocity on the order of 2.5 m/s, this guarantees that the loss of pair is on the order of 30%.

Before processing, individual PIV views are extracted from the images and subsequently warped to correct for distortion. The final PIV images are of size 550 W × 950 H  pixels. Registration shift that results from a misalignment between the laser sheet and the calibration target can be corrected by analyzing the raw SPIV images. Seeds that are bonded together create unusual bright pixels that can be identified on both views. Averaging the difference of the pixels location on each views yields the registration shift within experimental uncertainties (Fig. 6). The present analysis assumes that the laser sheet and the calibration target planes are parallel, but could be extended to the more general case where they are not parallel. In the present study, an average of 200 such pixel patches were systematically identified to correct for registration shift.

The PIV processing algorithm dynamically adjusts the interrogation window size at each PIV passes if correlation can be found. The windows are hence set between 64 × 64 pixels and 8 × 8 pixels from an initial value of 32 × 32  pixels. In this test case, 90% of the final windows are 16 × 16 pixels and 32 × 16 pixels. The windows steps are kept fixed at 8 pixels in both directions, which is about 140 μm. Finally, the vector fields are spectrally filtered with a long-pass filter to remove outliers. The PIV processing algorithm is described in great details in Sholl and Savaş (1997).

The combination of sharp cutoff filters and fluorescent particles proved effective in blocking glare and reflections from the free surface, Fig. 4. However, the systematic identification of this interface based on SPIV images only is rather challenging due to stereoscopic vision, finite thickness of the laser sheet, and reflected light from the free surface. In particular, the reflections can lead to spurious measurements.

The identification and reconstruction of an edge in stereoscopic imaging is a complex problem. Identifying a physical point on two images can be ambiguous and at times impossible. For example, a convex edge can be optically occlusive for one view and not the other (Faugeras 1993). Interface location recorded in either view will be displaced with respect to each other by an amount depending on the incidence angle and proportional to the laser sheet thickness.

At the interface between two media of different refractive indices, light is either reflected or transmitted or both. Snell’s equations give the reflection and refraction angles, while Fresnel’s laws give intensity of s- and p-polarized components of the reflected and transmitted beams (Hecht 1998). The reflectance and transmittance for the oil-air interface are shown in Fig. 7. For incident angles smaller than 40°, most of the light is transmitted through the interface and very little is reflected. For the most unfavorable case at the Brewster or polarization angle, θ _p  = 34.6°, the interface does not reflect any p-polarized light, and the s-polarized reflected light intensity is less than 13%. For incident angles larger than the critical angle, θ _c  = 43.6°, there is total internal reflection.

Since the viewing angles in Fig. 3 (α₁ = 43.0° and α₁ = 41.8°) are close to the Brewster angle (θ _p  = 34.6°), the interface does not act as a mirror for the light scattered by the particles and regions outside the layer appear dark (Lin and Perlin 1998). In this case, the incident angle, θ _i , employed in Fig. 7 is the angle between the normal to the free surface and the viewing angle. However, the strong free surface irregularities (see Fig. 2b), in particular troughs, locally increase the viewing angle with respect to the interface and hence act as mirrors for the fluorescent light (Fig. 8a). That is the source of the ligaments like patterns observed on the outside of the layer of view # 2 in Fig. 9.

When the free surface forms a small angle with the incident laser sheet (laser sheet nearly normal to the interface), most of the laser light is transmitted and very little is reflected (Fig. 7, here the incident angle is taken between the incident laser sheet and the free surface normal). This is the most favorable case to conduct PIV measurements in a plane perpendicular to the average free surface location. As the degree of waviness increases, the optical geometry rapidly shifts away from this condition. In certain cases, the angle formed between the incident laser light and the free surface is such that locally total internal reflection occurs making the interface virtually disappear (Fig. 8b). This particular case renders the identification of the free surface position on the PIV images difficult. In addition for higher injection flow rates than the one treated in this paper, drops formation and bubble entrapment contribute to blurring the interface. As a result, an edge finder algorithm that systematically tracts the liquid-air interface has not been successfully developed. A sample picture showing the out of plane particles that blur the interface position are presented in Fig. 9.

Total internal reflection at the interface also has a negative effect on flow field measurements. As illustrated in Fig. 8c, the laser sheet can be redirected inside the layer in a plane, nearly parallel to the incident sheet that can appear in focus on the object plane. Hence two planes are, at times, simultaneously recorded and manifest themselves in two ways on the data. Most frequently, this results in a local increase of background intensity and a decrease of the signal to noise ratio. However, in rare events when the liquid surface has both large axial curvature and small radial curvature, discernable out of plane seeds appear on the PIV data and lead to erroneous data. This last phenomenon could be tentatively tracked by monitoring the seed density of the PIV images and discarding SPIV samples with a statistically abnormal high seeds density. The authors have not implemented this technique as this phenomenon is very localized and appears in less than 5% of the frames. It should be also noted that these phenomena are minimized if the Scheimpflug condition is satisfied (the object and investigation planes coincide) and a fast lens is employed.

Uncertainties in turbulent flow measurements are due to systematic errors of instruments and convergence of statistics. The measurement errors in our systems are mainly due to PIV processing algorithm (aLPT), misalignment of calibration target with laser sheet, and camera calibration uncertainties.

For the error analysis and the presentation of subsequent data, it is convenient to decompose the instantaneous velocity, \({\mathcal{U}}_i,\) into an ensemble averaged portion, \(U_i \equiv <{\mathcal{U}}_i>\!\!,\) and a fluctuating portion, u _i , following the Reynolds decomposition, \({\mathcal{U}}_i = U_i + u_i.\) For the rms values of individual fluctuation quantities, we use \(u_i' = \sqrt{< u_i u_i >}\) where no summation over index i is implied. The ensemble averaging <   > is done over 1024 independent PIV samples.

Sholl and Savaş (1997) estimated the accuracy of aLPT to be 1% of the measured velocity. The misalignment of the laser sheet and the calibration target planes are calculated analytically to have a maximal 0.5% effect on the velocity measurements. Calibration errors of camera views are estimated by varying the location of the calibration reference points and shifting the views with respect to each other, both by ±1 pixel (i.e. the calibration and dewarping precision). The new raw images are then dewarped and interrogated to extract the three velocity components. The percent error between the modified and original data is computed for the mean and rms velocities. The errors for \({\mathcal{U}}_r, {\mathcal{U}}_\theta,\) and \({\mathcal{U}}_z\) are respectively 3.0, 3.0, 1.5% on the mean and respectively 3.0, 3.0, 1.0% on the rms.

The statistics convergence is estimated following methodology for stationary processes with Gaussian distribution and u _i ′ standard deviation around the mean, Baird (1995). The standard deviation on the mean and on the standard deviation are respectively \(u'_i/\sqrt{N}\) and \(u'_i/\sqrt{2(N-1)},\) with N the number of samples (i.e. 1024). Thus, the standard deviation on U _r is 5%, while it is only 0.3 and 0.4% for U _θ and U _z . The large uncertainty for U _r is due to the fact that \(\mid U_r \mid \) is smaller than its rms. For the rms velocity, the standard deviations are on the order of 2.5% for u _r ′ and 2.0% both for u _θ′ and u _z ′. 95% confidence interval is twice the standard deviations and is used to compare the instrumental and statistical precisions.

Finally, the combined uncertainty of instruments and statistics is estimated as the maximum of all errors for each component of the mean and rms velocities. Calibration uncertainties limit the accuracy of U _θ and U _z , while U _r , u _r ′, u _θ′, and u _z ′ are limited by statistics.

In the data presented below, a band of four velocity vectors have been masked around the edges of the flow fields to remove spurious vectors resulting from PIV processing near the edges of the images. The usable flow field has then 39 W × 107 H vectors. Further, the free surface is not systematically identified for reasons explained in Sect. 5.2 as a result this interface is masked as well.

In the experiments, PIV data are recorded at five stations for three injection flow rates and at three oil temperatures. The data presented here are acquired at about 2.4 tube diameters from the injection nozzle, with 2.1 × 10⁻³m³/s flow rate, and 25°C temperature. For this flow, the layer thickness is about 7.5 mm. The swirl ratio is about 5/3. Due to the particulars of the construction of the glass viewing prism, the flow in the immediate vicinity of the tube wall is not accessible. Thus, no reference to viscous sublayer is made.

The mean flow Reynolds number based on the average speed and the layer thickness is 3500. The (longitudinal) Taylor microscaleis about 700 μm, 5 times the PIV interrogation step size. The Taylor Reynolds number, Re_λ = u′ λ/ν, based on the Taylor microscale, and the rms velocity \({u'} = \sqrt{{\frac{1}{3}} < u_i u_i> }\) with summation over index, is about 70 for the dataset presented here.

For reasons covered in Sect. 5.2, the free surface could not be systematically resolved from the SPIV images, instead external instantaneous pictures, such as Fig. 1b, are employed to estimate the amplitude of waves at the liquid-air interface. For the particular flow conditions presented here, no drops were generated and there are no evidence of gas bubbles entrapment; the free surface irregularities variance is 4.5 ± 1.2% of the layer thickness.

That translates to waves with 1 mm rms amplitude for the prototype. This small amplitude is well below the 5 mm standoff distance between the heavy-ion beam and the liquid vortex for the heavy-ion fusion point design (Yu et al. 2003) insuring that the liquid layer will not interact with the ion beam. The current design can thus be employed for neutron shielding.