Bichromatic particle streak velocimetry bPSV
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Abstract
We propose a novel technique for threedimensional threecomponent (3D3C) interfacial flow measurement. It is based on the particle streak velocimetry principle. A relatively long integration time of the camera is used for capturing the movement of tracer particles as streaks on the sensor. The velocity along these streaks is extracted by periodically changing the illumination using a known pattern. A dye with different absorption characteristics in two distinct wavelengths is used to color the fluid. The depth of particles relative to the fluid interface can then be computed from their intensities when illuminated with light sources at those two different wavelengths. Hence, from our approach, a bichromatic, periodical illumination together with an image processing routine for precisely extracting particle streak features is used for measuring 3D3C fluid flow with a single camera. The technique is applied to measuring turbulent Rayleigh–Bénard convection at the free airwater interface. Using Lagrangian statistics, we are able to demonstrate a clear transition from the Batchelor regime to the Richardson regime, both of which were postulated for isotropic turbulence. The relative error of the velocity extraction of our new technique was found to be below 0.5 %.
Keywords
Particle Image Velocimetry Empirical Mode Decomposition Particle Tracking Velocimetry Hilbert Huang Transform Turbulent Flow Field1 Introduction
Interfacial transport processes of momentum, heat and mass across fluid boundary layers are of major importance in many industrial applications, environmental processes and fluid dynamic research. Since turbulent transport in the boundary layer of an interface plays a key role in the understanding of these processes, many investigators focused on the theoretical description of turbulent flow fields in the presence of an interface as well as on the development of numerical and experimental techniques. Although all these efforts have spawned a detailed theoretical description, for example, the advectiondiffusion equation and the NavierStokes equation, the physical workings are far from being completely understood. The interdisciplinary interest in the said processes is shown by publications in biophysical research (Berthe et al. 2010); Kertzscher et al. 1999), engineering (Burgmann et al. 2008) and environmental physics (Banner and Peirson 1998). Main challenges associated with measuring flow information close to an interface are the inherent threedimensionality of turbulence that implies the measurement of volumetric threedimensional information and the coupling of obtained flow information with the position of the interface.

laser speckle velocimetry (LSV), which assesses flow information by analyzing random interference patterns caused by a high seeding density of tracer particles,

particle tracking velocimetry (PTV), which uses lower seeding densities to enable a tracking of single particles through a long exposure time or multiple images in a sequence and

particle image velocimetry (PIV), which extracts the flow information from a fluid with a medial seeding density where the displacements of small groups of particles within an interrogation window are analyzed statistically.
Most proposed approaches in third category use correlationbased methods to find similar particle constellations within an interrogation window in consecutive image pairs. This mode is often referred to as “standard PIV.” In the recent history of PIV, there were multiple attempts to obtain 3D3C data. These approaches can also be distinguished according to the technique used to obtain 3Dvelocity information and the methods used to measure volumetric datasets. The most popular method to access three components (3C) of the velocity field is to measure the outofplane velocity using stereoscopic methods (Prasad 2000), extended by a technique called multiplane PIV by using multiple lasersheets (Kähler and Kompenhans 2000; Müller et al. 2001; Liberzon et al. 2004; Cenedese and Paglialunga 1989; Brücker 1996) or intensitygraded light sheets (Dinkelacker et al. 1992) for the reconstruction of the third velocity component. In the recent past, many methods were proposed to access volumetric information from 3D flow fields. This was achieved using holographic measurements (Hinsch 2002; Sheng et al. 2008), by combining PIV with Doppler global velocimetry (PIV/DGV) (Wernet 2004), using tomography (Elsinga et al. 2006; Schröder et al. 2008), defocusingbased approaches (Pereira et al. 2000, 2007; Willert and Gharib 1992), scanninglightsheet methods (Burgmann et al. 2008; Hoyer et al. 2005; Brücker 1995), colorcoded depth resolving techniques (Zibret et al. 2003; McGregor et al. 2007) and absorptionbased methods (Jehle and Jähne, 2008; Berthe et al. 2010).
To test the validity and applicability of this new method, we applied it to a semiartificial test sequence with highly accurate ground truth from (Berthe et al. 2010) and to the measurement of buoyancydriven, turbulent Rayleigh–Bénard convection. The properties of this type of turbulent flow are well understood and make an assessment of the statistical properties derived from our technique feasible. Also, due to the long characteristic time constant, a long integration of our measurements is necessary to derive valid statistics. Errors in the measurements would thus add up and become apparent. All this makes this type of setting an ideal verification experiment for the newly proposed technique. The turbulent flow is analyzed by applying Lagrangian pair dispersion statistics. Our measurements clearly show a transition from a t ^{2} scaling in the Batchelor regime to a t ^{3} behavior as predicted by the Richardson law as described in (Salazar and Collins 2009).
1.1 Interfacial measurement technique
For the understanding of interfacial transport processes, a detailed knowledge of flow mechanisms close to the boundary is of fundamental importance. The need of collocated measurements of fluid flow within the thin interfacial boundary layer and its topography poses a major challenge, as summarized by Turney et al. (2009). Especially for curved and/or dynamically moving interfaces, flow fields are often highly complex. This necessitates volumetric 3D measurement techniques which extract flow features in a Lagrangian frame of reference. Moreover, for most analyses the motion relative to the interface is sought. To this end, all stereoscopic, holographic and tomographic methods need to be extended by an additional interface tracking approach to allow for a coordinate transform between the inherent fixed laboratory reference frame and the moving interface (Banner and Peirson 1998; Turney et al. 2009; Law et al. 1999). The standard approach for interface tracking in 2D measurements is usually achieved by employing an additional camera to observe the intersection line between a lasersheet and the interface. This additional measurement in conjunction with cross calibration issues may lead to significant sources of error. This problem becomes highly challenging for tracking 2D topographies, as required by volumetric measurements.
Absorptionbased depth estimation methods, as proposed by Debaene et al (2005), present a promising alternative which circumvents these experimental difficulties. In this approach, the distance away from the interface is encoded by the particle’s intensity according to LambertBeer’s law. Other depth from attenuation methods also uses the imaged intensity in a similar way to assess information from interfacial regions (Liu and Sullivan 1998). The technique that is presented in this paper is an extension of current absorptionbased techniques. To test its validity and applicability, it is applied to the free airwater interface. This scenario represents a challenging, yet highly relevant application with a moving interface and a thin boundary layer that has a vast influence on the interfacial exchange of heat, mass and momentum.
Refraction at curved interfaces presents a severe obstacle for most interfacial measurement techniques. In the presented study, curved interfaces inhibit a precise measurement for two reasons. Firstly, the determination of the tracer particle position can be corrupted by a curved interface since particles are imaged through the interface. Secondly, the presented method relies on a homogeneous illumination which can be distorted by focusing effects. For some interfaces these refraction artifacts can be avoided by using refractive index matching methods (Budwig 1994).
1.2 Bichromatic depth estimation approach
The development of a dualwavelength method for absorptionbased depth estimation was motivated by a method used in biofluidmechanics (Debaene et al. 2005) where the exponential absorption characteristic of a dye was used to extract the depth of reflecting tracer particles in a fluid. However, the grayvalue of an imaged tracer particle does not only depend on its depth as given by LambertBeer’s law. It also depends on their reflectance properties. This is mainly dominated by the size of the particle. In Debaene et al. (2005), great care was taken to use only particles of identical size. However, this is impractical for a number of applications and also error prone.
1.3 Particle streak velocimetry
As shown in recent publications (Sellent et al. 2010), the use of timeintegrated, longexposure images to obtain information about moving objects has become an important topic in image processing. Particle streak velocimetry (PSV) is often also referred to as Streak Photography or Particle Streak Tracking (PST). In PSV long exposure times are used to map particle trajectories by a temporal integration in the image plane. To the best of our knowledge, this method was first introduced by Fage and Townend (1932) in a study to visualize characteristics of turbulent flow fields in circular and rectangular pipes. Afterward, streak photometry was also used to visualize flow characteristics by Prandtl (1957). First measurements to obtain quantitative information from particle streak images were conducted by Dimotakis et al. (1981), Dickey et al. (1984) and Adamczyk and Rimai (1988). All three studies used computerized evaluation routines to extract the mean direction and the mean velocity by subtracting the endpoints of each streak. The study by Wung and Tseng (1992) can be seen as an early precursor of the technique presented in this paper since this is the first time that temporal information was coded along the streak structure by changing the intensity of the illumination during exposure.
In the recent past many approaches were introduced to extend this method that was originally developed to measure two velocity components in a plane (2D2C), to measure a third velocity component (Wung and Tseng 1992; Müller et al. 2001) or volumetric data (Sinha and Kuhlman 1992; Rosenstiel and RolfRainer Grigat 2010; Biwole et al. 2009; Dixon et al. 2011). A newly proposed method, published in Dixon et al. (2011), even uses a holographic PSV technique to measure volumetric flow features. In this approach blurred holograms are recorded by imaging particles that move during the exposure time. From the radial intensity profile of the particles in the hologram, the magnitude and direction of the inplane velocity can be computed.
Another general shortcoming of PSV and PTV methods is the low spatial sampling rate that is introduced by a systematic restriction to low seeding densities. Nonetheless, due to the subpixel precise centerline extraction and the frequencybased velocity extraction, the sparse data comprise a high spatial and temporal precision that allows for accurate statistical analyses of the Lagrangian velocity information.
1.4 Bichromatic extension to the standard PSV approach
The method presented in this paper can be seen as an extension of the standard 2D PSV as described in (Adrian 1991), by a bichromatic depth estimation approach as introduced by Jehle and Jähne (2008) and a frequencybased velocity extraction. Moving particles are illuminated with an alternating twowavelength illumination that is used to code velocity information in the longexposure images. In contrast to all existing methods, we did not solve the extraction of single streaks by a thresholdbased segmentation followed some fitting routine (Müller et al. 2001; Rosenstiel and RolfRainer Grigat 2010), but implemented an tensorbased, iterative centerline extraction routine (described in Sect. 3). Advantage of this extraction strategy is that it enables a subpixel precise extraction of the trajectory’s centerline and intensity. The method also incorporates additional knowledge by assuming a Gaussian intensity profile perpendicular to the streak as a consequence of (6). Since we used an alternating bichromatic illumination and an absorbing dye, the intensity signal along the streak can be used to compute the distance between particle and interface for all points on the particles trajectory using (2).
Instantaneous velocity extraction
2 Measurement setup
2.1 Calibration and measurement
The accuracy of the depth extraction using (2) depends strongly on a precise estimation of the penetration depths z _{*i } and initial intensity ratio \(\frac{I_{02}}{I_{01}}\) for both wavelengths \(\lambda_i\ i\in{1,2}\) (Berthe et al. 2010). This makes a calibration measurement of the exact extinction behavior of the dye for the given concentration and wavelengths indispensable to guarantee a high accuracy of the approach.
In all measurements of this study we used a white, diffuse target that was placed in the measurement volume at a defined angle to the optical axis and illuminated the calibration target with both wavelengths separately. Using these calibration measurements we extracted the initial intensity ratio from the part of the calibration target that was above the surface. The penetration depths for both wavelengths were determined by an exponential fit on the grayvalues along the target.
3 Feature extraction
The extraction of the flow information can be separated into a set of subtasks. Each image from a recorded sequence is preprocessed followed by the centerline extraction and the depth and velocity estimation. In the last step the direction ambiguity of the velocity is solved using a matching algorithm, successively comparing each image with its predecessor and its successor. The resulting volumetric, Lagrangian trajectories describe the paths of single tracer particles over several images.
3.1 Boundary tensor
3.2 Iterative centerline extraction
 1.
Initialize the streak width with σ = 1, startPos = current position
 2.
Extract the grayvalues from a crosssectional line b with elements \(b_i, i=1\dots N\) perpendicular to the streak structure (the direction of the streak structure is known from the local orientation θ computed using the boundary tensor).
 3.
Use a nonlinear LevenbergMarquard (LM) fit to estimate the offset A, the amplitude B and the width σ of a Gaussian bellcurve \(g(x) = A + B * {\text{exp}}(\frac{(x \mu)^2}{2 \sigma^2})\) on the grayvalue profile along b. The initial guess for the parameters is: \(\tilde{\mu} = N/2; \tilde{A} = \min(b_i, \forall i=1\dots N); \tilde{B} = \max(b_i, \forall i=1\dots N)A. \)
 4.
The subpixel precise position of the CL is computed from the current pixel position and the position of the maximum μ of the Gaussian bellcurve. Since the LM fit considers multiple intensity values and a model of the intensity course of the scattered light, the amplitude B provides a much more reliable information on the streaks intensity at the position of the centerline.
 5.
IF σ < threshold THEN Shift the current position by a predefined step sizes s = (0,1][px] in the direction of the local orientation and go to step 2 ELSE Go to the startPos and travel along the streak in the reverse direction until the width σ is larger than a certain threshold.
The result is a list of subpixel precise coordinates of the CL position and the intensities at these positions. The achievable precision of the horizontal trajectory position is therefore better than the pixelresolution of the object space imaged by the camera.
3.3 Frequency analysis
Since the frequency of the periodic intensity modulation of the alternating twowavelength illumination can be set precisely to a constant value, it is possible to estimate the horizontal velocity of a particle from the spatial intensity signal along the corresponding streak structure in the image.
In the field of signal processing, many attempts to extract signal frequencies with the highest possible precision in the frequency and temporal domains were made. The most promising approaches are the Windowed Fourier Transform (sometimes also called short time Fourier Transform (STFT) (Gabor 1946), the Hilbert Huang Transform (HHT), which can be used to extract an instantaneous frequency and a instantaneous amplitude of the input signal (Smith 2007), and the WignerVille Transform, which can be seen as a method for measuring the local timefrequency energy (Martin and Flandrin 1985). In order to obtain an instantaneous frequency information that enables the extraction of the particle velocity at every point along the streak, we perform a HHT on the intensity signal extracted along the streaks CL. Additionally, we make use of the instantaneous amplitude signal, since it contains information on the particle depth (cf. Fig. 2).
3.4 Hilbert transform
3.5 Depth reconstruction
3.6 Construction of lagrangian trajectories
To solve the problem of directional ambiguity, described in Sect. 1, we make use of information from the previous and the following bPSV image in a sequence. We track the trajectories of single particles through multiple exposures of an image sequence in an iterative algorithm. The problem of finding corresponding streaks in subsequent frames can be a challenging task (Rosenstiel and RolfRainer Grigat 2010). In this paper, we propose the use of a heterogeneous Mahalanobis distance measure (17) to track a particle along its trajectory through a set of images. The width σ and the orientation at the streak end, extracted from the local orientation θ, are used to define a proper distance measure and threshold a feasible trajectory matching between consecutive frames. This allows us to successfully perform a robust matching in most cases.
After the construction of each Lagrangian trajectory, we use the horizontal velocity from (20) for a selfvalidation. By integrating \(\tilde{v}_h(t)\) over time, we compare the obtained trajectory length with the sum over the lengths of all streaks corresponding to the trajectory. A large difference between these two measures indicates an error in the velocity extraction or an incomplete trajectory. For the following experiments, we allow a maximal deviation of 5 %, all trajectories that are above this threshold are rejected.
3.7 Lagrangian twoparticle dispersion
For the computation of Lagrangian particle pairs, we analyzed the whole set of trajectories that results from a image sequence. Out of all possible trajectory pairs, we extracted those which had a distance \({\tilde{\bf x}}_1(t_p){\tilde{\bf x}_2}(t_p) =r_0\) at some point in time t _{ p }. From this point in time until the trajectory pair not longer coexists, this particle pair contributes to the Lagrangian average \(\langle.\rangle_L\) in (22).
As mentioned by Ott and Mann (2000), it is desirable to choose a small initial separation r _{0} which introduces a dilemma, because choosing a smaller r _{0} means reducing the number of valid particle pairs and causes a smaller sample size in the Lagrangian statistic. For the Lagrangian twoparticle statistics extracted in the course of this study, we computed the pair dispersion for initial distances within η_{ K } ≪ r _{0} ≪ 10η_{ k }.
4 Experiments
4.1 Benchmark datasets
Statistical evaluation of the benchmark results. The deviation of the mean velocity \(\overline{v(x)}\) from the ground truth given by the speed of the micrometer traverse is less than one sigma of the standard deviation
Measurement  Velocity \([\frac{\hbox{mm}}{\hbox{s}}]\)  F [Hz]  \(\overline{v(x)}\) (mm s^{−1})  \( {\text{std}}\left(v(x)\right)\) 

#1  25.0  0.2–2.0  24.98  0.09 
#2  9.1  0.2–1.4  9.12  0.04 
#3  9.1  0.2–1.4  9.08  0.04 
#4  9.1  0.2–1.4  9.09  0.04 
#5  9.1  0.2–1.4  9.11  0.03 
The distributions of the datasets #2 to #5 are shown in a boxandwhisker diagram provided with the supplementary data of this paper.
4.2 Rayleigh–Bénard convection experiment
To prove the applicability of the proposed technique in a realworld measurement, bPSV datasets in the watersided boundary layer of a turbulent Rayleigh–Bénard (RB) convection were recorded. In the field of fluid mechanics, convectiondriven turbulent flows belong to the beststudied flow fields. In various studies RBconvection was analyzed based on Eulerian (Lohse and Xia 2010) and Lagrangian statistics (Schumacher 2009; Emran and Schumacher 2010; Sreenivasan and Schumacher 2009; Gasteuil et al. 2007). Since bPSV measurements provide densely sampled, Lagrangian information about position and velocity, we focus on a Lagrangian interpretation of the measured data.
Experimental setup.
For the generation of a welldefined turbulent RBconvection, a rectangular vessel with the dimensions (H = 147mm, L = 405mm, W = 415mm) was built from 3.3mmthick BOROFLOAT^{®} glas. For the heating we mounted a mirrored box with two infrared heating tubes (Heraeus Noblelight, Art.Nr.: 45132877) with P _{max} = 1kW and \(T_{{{\text{max}}}} = 1.2 \times 10^{3} \;^{^\circ } {\text{C}}\) each under the vessel. A sketch of the experimental setup is shown in Fig. 8. Due to the high transmission of 93 % at wavelengths between 2.0 and 2.7 nm, most of the infrared light penetrates the bottom plate of the vessel and is absorbed within the first millimeter of the water. The cooling at the water surface is mostly due to evaporation and radiation. In all experiments the vessel was filled up to a height of \(\tilde{H} = 50\hbox{mm}\) and the heating power was set to 945W. Measurements were taken once the system reached its state of equilibrium at a water temperature of T≈51°C. The temperature difference \(\Updelta T=23.6\,^\circ\hbox{C}\pm 0.3\,^\circ\hbox{C}\) between the bottom plate and the air directly above the water surface was measured by a Pt100 thermo sensor using a GMH 3710 thermometer (GREISINGER electronic GmbH). All temperatures were averaged over several minutes. The absorbing dye Tartrazine was added to the liquid from a stock solution with 1g Tartrazine per liter, to obtain a concentration of 12 mg l^{−1} Tartrazine in the water. This concentration enables a particle extraction down to the depth of 10mm below the interface. Every pixel of the 5Mpx sensor imaged a square of 30 μm x 30 μm in the interfacial region. From the pixel size and the penetration depth, the spatial uncertainty of the method can be approximated to be \(\Updelta x = \Updelta y \leq 30\,\mu\hbox{m}\) and \(\Updelta z \leq 500\,\mu\hbox{m}\). For the seeding we used neutrally buoyant silvercoated hollow ceramic spheres with a mean diameter of 100 μm and a mean density of 0.9 ± 0.3 g cm^{3} (Potters Industries Inc. ConductOFil^{®} AGSL15016TRD). These tracer particles were sorted in advance based on their sedimentation behavior in water to obtain a narrower density distribution. During the experiment, the experimental setup was covered with lightabsorbing material to ensure that the light that is reflected by the particles only originates from the LED arrays used.
An analysis of the extracted particle velocity distributions and the acceleration distributions showed no evidence for a corrupting influence of the previously mentioned systematic bias toward high velocities. Nonetheless, this special topic is to be addressed in further studies.
5 Conclusion
The bichromatic particle streak velocimetry (bPSV) method represents a highly accurate and simpletouse measurement technique for extracting Lagrangian 3D3C flow information in turbulent interfacial flow fields. This new measurement technique relies only on one camera and extracts particle positions relative to the fluid interface. Horizontal position and velocity of the particle trajectories are extracted from longexposure images using a harmonically modulated illumination to code temporal information along the recorded particle streaks. The vertical information is obtained from a bichromatic depth extraction using an absorbing dye and a dualwavelength illumination. The approach is highly robust to a wide range of encountered velocities and illumination frequencies. This was shown on a previously published PIV benchmark dataset (Berthe et al. 2010). The deviation of the extracted mean velocity from the ground truth lies within the standard deviation of lower than 0.5 % of the velocity.
Also, the interfacial flow field of a turbulent Rayleigh–Bénard convection was analyzed using bPSV. In this regime, the turbulence is highly anisotropic. Previously, only limited experimental data existed for this type of forced turbulence. Pair dispersion rates for different inertial pair separations were extracted from the measured Lagrangian trajectories. In the Lagrangian statistics, a clear transition from an inertial subrange which corresponds to a Batchelor regime with a t ^{2} scaling behavior to a t ^{3} in the Richardson regime was found. Both regimes were originally defined for isotropic turbulence. Our studies confirm the applicability of this theory in the anisotropic case as well. These results clearly demonstrate the accuracy and applicability of bPSV for research in interfacial flows. Currently, the only limitation of the approach arises at timedependent, freely moving interfaces due to refraction processes taking place. In a next step, we will address these issues. bPSV will thus become a “new stone in the wall” for fluid flow measurements in challenging interfacial flows.
Notes
Acknowledgments
We gratefully acknowledge the financial support by the DFG Graduate College GRK1114 “Optische Messtechniken für die Charakterisierung von Transportprozessen an Grenzflächen.”
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Supplementary material
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