Error quantification of 3D homogeneous and isotropic turbulence measurements using 2D PIV
Introduction
PIV has been utilized to measure various flows, ranging from basic shear flows, pipe and channel flows, to supersonic flows. Although many of these flows are inherently three-dimensional in nature, 2D PIV is still widely used due to its simple set up and high reliability (Scharnowski et al., 2017a, 2017b). However, when measuring 3D flows with this 2D technique, errors can occur due to out-of-plane motion. Out-of-plane motion occurs when the tracer particles move in a normal direction to the image plane and disappear. Errors in the measured velocity field will propagate to the turbulence statistics, producing larger errors and consequently distorting turbulence characteristics. Thus, it is important to properly understand 3D flow effects before conducting experiments.
PIV error due to out-of-plane motion has been analyzed previously. Keane and Adrian (1990) analyzed the degradation of correlation peaks by out-of-plane motion through Monte-Carlo simulations and optimized system parameters. They noted that relative out-of-plane displacements of less than 40% may have a valid particle pair detection probability of more than 90% at sufficient particle image density and proper in-plane displacement. Hart (2000) showed that when out-of-plane displacement increases, the probability of valid correlation decreases by up to nearly 30%. He showed that the use of correlation based correction (CBC) can significantly reduce the number of error vectors. It should be noted that these studies focused only on the correlation signal magnitude and error vector detection probability.
There have been attempts to quantify the out-of-plane motion. If the particles disappear from the image due to out-of-plane motion, the matching particle pairs cannot be found between the two consecutive images. This is called loss-of-pair. Keane and Adrian (1992) quantified the loss-of-pair effect through the Fo factor. The Fo factor is a measure of the degree to which particles disappear from the focal plane, based on the degree of intensity change when the laser intensity profile and out-of-plane direction displacement are known. It was defined to be Fo = 0 when all the particles lose pairs between the two images by out-of-plane motion, and Fo = 1 when there was no loss of pairs. The Fo value is highly correlated with error because when it has a low value, the particle pair detectability and signal to noise ratio (SNR) of valid vectors decreases during the cross-correlation process. Keane and Adrian, 1990, Keane and Adrian, 1991 selected three factors that affect PIV detectability: Fo, particle image density (NI), and the normalized correlation of the interrogation intensity across the interrogation window (FI), and grouped them into one indicator (NIFIFo). They created a 'design rule' such that NIFIFo ≥ 7 (Westerweel, 1997), and this criterion has been considered the rule of thumb for many PIV studies. Scharnowski and Kähler (2016) quantified particle displacement error by using this Fo factor and the volume ratio between auto-correlation and cross-correlation. Originally, the Fo factor was difficult to obtain in experiments, because it is necessary to know the out-of-plane direction displacement. This method makes it possible to calculate Fo using images obtained from experiments. They used synthetic image tests with experiments to take advantage of the ability to precisely control all PIV parameters in a range that is difficult to implement by experimentation. They estimated the displacement vector uncertainty with respect to the Fo value, and optimized this value to minimize the uncertainty. All of these studies have used Fo to assess displacement detectability, uncertainty, and error.
The error and uncertainty of the displacement caused by the out-of-plane motion propagates to velocity error. This, in turn, can have a large effect on turbulence statistics such as turbulence kinetic energy or viscous dissipation rate, which are defined using velocity fluctuations and their spatial derivatives. Wilson and Smith (2013b) assessed the propagation of local instantaneous uncertainty from PIV results. They estimated the uncertainty of mean velocity and Reynolds stresses from the velocity uncertainty, due to four parameters: particle image displacement, particle image diameter, particle number density, and the effect of shear. The uncertainty was calculated using the Taylor series method proposed by Wilson and Smith (2013a), and validated through experiments. Although they successfully estimated uncertainty for Reynolds stresses, they did not estimate the uncertainty for the dissipation rate, and Taylor or Kolmogorov length scales, which are important statistics in turbulence. In addition, the experiment did not include uncertainty estimation for out-of-plane motion, which can be a significant error source in 2D PIV.
The flow field we are specifically interested in is homogeneous and isotropic turbulence (HIT), which has a significant amount of 3D motion. HIT is the most ideal type of turbulent flow, which can be used to examine basic turbulence theory. There have been many efforts to create HIT in the laboratory. Wind tunnels were initially used to generate turbulence via grids, and HIT was observed within downstream planes. However, the turbulence decays downstream of the grid, and there is a mean flow superposed on the turbulence fluctuations. Birouk et al. (1996) and Fallon and Rogers (2002) implemented HIT with no mean flow in a confined box using rotating fans. Hwang and Eaton (2004) utilized synthetic jet actuators with subwoofers to create a similar flow, and were able to achieve a higher Taylor microscale Reynolds number of 218. They used standard 2D PIV to characterize the flow. Homogeneity and isotropy were quantified using root-mean-square (RMS) velocity fields. Turbulence statistics including turbulent kinetic energy (TKE), dissipation rate, and energy spectrum were calculated.
Many following studies have attempted to generate HIT using a similar approach as Hwang and Eaton (2004), and analysis has been conducted via various measurement techniques. Zimmermann et al. (2010) obtained the structure function through Lagrangian particle tracking. Using the relationship between the structure function and turbulence statistics such as dissipation rate, Taylor Reynolds number and homogeneity were calculated. The experimental setup was very complicated because three cameras were used. Chang et al. (2012) used laser Doppler velocimetry (LDV) to calculate the structure function by measuring the velocity of two adjacent points in the middle of the instrument. Goepfert et al. (2010) combined 2D PIV with 2 components LDV at two different points in a non-confined experimental setup. They calculated the turbulence statistics by using velocity fields from 2D PIV, assuming that the 3D flow is homogeneous and isotropic as in Hwang and Eaton (2004). Similarly, De Jong et al. (2009) obtained the velocity field in an apparatus with mounted fans using 2D PIV, and calculated the dissipation rate using various methods. Dou et al. (2016) constructed a two-scale PIV technique which consisted of two independent PIV systems to capture the same area at different sizes in a truncated icosahedron-type apparatus. They point out that there is a limit to obtaining turbulence statistics using point measurement techniques such as LDV, which cannot simultaneously measure the spatial covariance of velocity over the region of interest (ROI).
Up to now, most studies have focused on the uncertainty and detectability of the velocity vector due to out-of-plane motion. Since turbulence statistics include first and second order derivatives and correlations, we aim to provide some insight on how the velocity error propagates. The main purpose of this study is to quantify the effect of out-of-plane motion on turbulence statistics measured with 2D PIV in HIT. Synthetic particle images are superimposed on forced isotropic turbulence DNS data, which allows straightforward control of many PIV parameters. Similar to the method of Hwang and Eaton (2004), various turbulence statistics will be calculated, and the 3D flow effects on these statistics will be assessed by comparing the DNS flow fields with the vector fields obtained via PIV. Parameters such as laser thickness, camera inter-frame time, and interrogation window size are optimized to reduce out-of-plane errors in the turbulence statistics. This information can be used to determine how much error results from 2D PIV when characterizing 3D flows such as HIT.
Section snippets
Synthetic particle image generation
Synthetic particle images are often used to quantify the performance of PIV algorithms and impact of various error sources, because the velocity fields of the moving particles are known, and various PIV parameters can be easily controlled (Kähler et al., 2016; Stanislas et al., 2003, Stanislas et al., 2005, Stanislas et al., 2008). Okamoto et al. (2000a, 2000b) developed a standard evaluation tool that can set various parameters to evaluate PIV algorithms. Parameters such as particle number
Results
Some turbulence statistics were selected to characterize the HIT, with reference to the study of Hwang and Eaton (2004). Turbulence kinetic energy (TKE), dissipation rate, Taylor microscale, Kolmogorov length and time scale, and velocity correlations are calculated from the DNS data and simulated PIV results. Before elaborating on the turbulence statistics, velocity error is first examined, to assess the synthetic images and PIV algorithm. This can provide a basic understanding of various error
Conclusion
In this study, the effect of out-of-plane motion on turbulence statistics measured by 2D planar PIV was investigated. Turbulence statistics respond more sensitively to out-of-plane motion than velocity because they contain fluctuating and derivative terms. The specific flow that was considered was 3D isotropic turbulence generated by DNS, from the Johns Hopkins turbulence database. Statistics such as turbulence kinetic energy (TKE), viscous dissipation rate, Taylor and Kolmogorov length scales,
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2017R1A2B4007372 and 2017R1A4A1015523). This work was also supported by the Research Resettlement Fund for the new faculty of Seoul National University.
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