As shown in recent publications (Sellent et al.
2010), the use of time-integrated, long-exposure 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 end-points 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 Rolf-Rainer 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 in-plane velocity can be computed.
Key idea of
particle streak velocimetry (PSV) is that tracer particles are imaged over a certain long exposure time
t
exp, instead of imaging the tracer in single-shot recordings with a very short integration time. The intensity distribution on the CCD-chip at a certain point in time
I(
x) is given by the well-known
Airy-Function, which in polar coordinates
\(\left(\rho, \varphi\right)\) is given by
$$ I\left(\rho,\varphi\right) = \left(\frac{2 J_1\left(k a \rho\right)}{k a \rho}\right)^2I_0. $$
(3)
Here
J
1 is the first-order Bessel Function,
\(k=\frac{2 \pi}{\lambda}\) is the wavenumber, and
a controls the radius. This function can be approximated by a Gaussian bell-curve as follows (Leue et al.
1996). The logarithm of the Airy-Function can be decomposed into a second-order Taylor expansion given by
$$ \ln\left(I(\rho,\varphi)\right)=\ln(I_0) -\left(\frac{ka\rho}{2}\right)^2+{\fancyscript{O}}(\rho^4). $$
(4)
By replacing
\(\sigma = \frac{\sqrt{2}}{k a},\) the first- and second-order terms can be rewritten to a Gaussian bell-curve in polar coordinates.
$$ I(\rho,\varphi)=I_0\exp\left(-\frac{1}{2} \left(\frac{\rho}{\sigma}\right)^2\right) $$
(5)
The gray-value distribution
g(
x) on the CCD-Chip, caused by the intensity distribution of a particle that travels along a trajectory
\({\mathbf{X}}(t),\,t \in (0,t_{{{\text{exp}}}}) \) during the exposure time, can therefore be modeled by a simple time integral
$$ g({\bf x})=\frac{1}{t_{exp}-t_0} \int_{t_0}^{t_{exp}}G_\sigma\left({\bf x}-{\bf X} (t)\right)dt. $$
(6)
where
\(G(\cdot)\) is a two-dimensional Gaussian distribution. For a set of
N tracer particles with trajectories
\({\mathbf{X}}_{l} (t),\,t \in (0,t_{{{\text{exp}}}})\) with
\(l=0,\ldots,N-1\) the gray-value distribution in a PSV image is given by the sum over all trajectories.
$$ g_{N} ({\mathbf{x}}) = \frac{1}{{t_{{{\text{exp}}}} - t_{0} }}\sum\limits_{{l = 0}}^{{N - 1}} {\int\limits_{{t_{0} }}^{{t_{{{\text{exp}}}} }} {G_{\sigma } } } \left( {{\mathbf{x}} - {\mathbf{X}}_{l} (t)} \right){\text{d}}t $$
(7)
The result is images that contain information on the particle trajectories recorded during the exposure time. Unfortunately, particles with a horizontal velocity below a certain threshold do not result in detectable streak patterns. To prevent a bias toward higher horizontal velocities, the optical axis in PSV measurements is commonly adjusted perpendicular to the principal direction of the observed flow field. In interfacial turbulence measurements, the optical axis should be directed perpendicular to boundary layer since pure vertical particle movements are unlikely to occur due to the anisotropic restriction of the flow. The aim of all PSV approaches is to extract flow features from these streak images by an approximation of the particle trajectories
\(\hat{\bf X}_N\). These trajectories can in a second step be used to extract Lagrangian fluid-flow features. Compared to the other volumetric 3D PIV approaches, this principle has a set of advantages. Rosenstiel and Rolf-Rainer Grigat (
2010) describe the advantages of PSV/PST compared with PIV (i.e., no limitation to narrow measurement areas of volumes, simple application to 3D3C measurements, less calibration effort, cost efficiency, simplicity, low seeding density). Main obstacle of PSV measurements is the loss of temporal information due to the time integral in (
6). The consequence is that the standard PSV approach enables the extraction of the particles’ average velocity during the exposure time interval (dividing streak length by exposure time), but the sign of the velocity remains unclear. This central restriction of PSV is called
direction ambiguity. In the recent development of PSV algorithms, several strategies were introduced to overcome this problem. A very intuitive solution for that problem as used, for example, by Scholzen and Moser (
1996) is to use an additional camera with a short exposure time to determine the streaks direction. Drawback is that by adding an additional camera one of the main advantages of PSV, the simplicity and the cost efficiency are not longer given. Adamczyk and Rimai (
1988), Müller et al. (
2001); Rosenstiel and Rolf-Rainer Grigat (
2010) proposed the use of an asymmetric timing pattern, strobed on the illumination signal to code the flow direction on the streak structures. Another advantage is that this method enables to check for particles that left the laser-sheet. On the other hand, strobing the illumination during one exposure creates a problem because the streak fractions generated by the strobing have to be matched correctly for the extraction of a velocity vector. A slightly different approach that does not need a matching of streak fractions was used by Wung and Tseng (
1992) who change the illumination intensity asymmetrically during the exposure. In our approach, intensity information codes velocity information. Therefore, we developed a matching algorithm that iteratively combines single PSV images from a sequence to trajectories by tracking particles through the image stack. This enables us to assign a direction to every streak that was detected in at least two subsequent images. As a by-product, this matching algorithm allows to track particles that comprise too slow horizontal movements within their trajectories. In these cases, lost information is reconstructed using the neighboring streaks.
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 sub-pixel precise centerline extraction and the frequency-based velocity extraction, the sparse data comprise a high spatial and temporal precision that allows for accurate statistical analyses of the Lagrangian velocity information.