On the noise in statistics of PIV measurements

William K. George

{}^{1}

Michel Stanislas

{}^{2}

Abstract

It is argued herein that when PIV is used to measure turbulence, it can be treated as a time-dependent signal. The ‘output’ velocity consists of three primary contributions: the time-dependent velocity, a noise arising from the quantization (or pixelization), and a noise contribution from the fact that the velocity is not uniform inside the interrogation volume. For both of the latter their variances depend inversely on the average number of particles or images in this interrogation volume. All three of these are spatially filtered by the finite extent of the interrogation window. Since the above noises are associated directly with the individual particles (or particle images), the noise between different realizations and different interrogation volumes is statistically independent.

${}^{1}$ Visiting Pr., Centrale Lille, F59651 Villeneuve d’Ascq, France
${}^{2}$ Pr. Emeritus, Centrale Lille, F59651 Villeneuve d’Ascq, France
corresponding author: [email protected]

List of Symbols

$\vec{a}$	Lagrangian particle coordinate
$b_{cam}(\vec{x}_{s},t)$	Particle image on detector
$B_{oi,j}(\vec{y},\vec{y^{\prime}})$	Two-points single-time velocity correlation tensor
$c(\vec{y},t)$	Image of all particles inside the interrogation volume
$C_{o}$	Correlation of images
$E_{11}(k_{1})$	One-dimensional velocity fluctuation spectrum
$E_{noise}(k_{1})$	Noise one-dimensional spectrum
$f(\vec{x},t)$	Particle image back-projected in physical space
$f$	Lens focal length
$f$ #	Lens f-number
$F_{oi,j}(\vec{k})$	Measured three-dimensional spectrum
$\tilde{F}_{i,j}(\vec{k})$	Volume-averaged three-dimensional spectrum
$F_{oi,j}^{1}(\vec{k_{1}})$	Measured one-dimensional spectrum
$\tilde{F}_{i,j}^{1}(\vec{k_{1}})$	Volume-averaged one-dimensional spectrum
$F_{i,j}^{1}(\vec{k_{1}})$	True one-dimensional spectrum
$F_{noise}^{1}(\vec{k_{1}})$	Noise one-dimensional spectrum
$g(\vec{a})$	Particle location function: $\delta(\vec{a})$
$I=1.492$	Integral over $[0,2\pi]$ of the sinc function
$\vec{k}$	Wave vector
$k_{i}$	Wave number along $x_{i}$
$M$	Magnification of recording system
$n(\vec{y},t)$	Number of particles inside the interrogation volume
$N(\vec{y})$	Average number of particles inside the interrogation volume
$N_{p}$	Number of particle per pixel (ppp)
$P_{i}$	Uncertainty on particle position due to pixelisation
$\vec{r}$	Position vector inside the interrogation volume
$t$	Time
$t_{o}$	Initial time
$U$	Mean velocity along $x$
$u_{i}(\vec{x},t)$	Instantaneous local velocity component
$u^{\prime}_{i}(\vec{x},t)$	Instantaneous local velocity fluctuation component
$\tilde{u}_{i}(\vec{x},t)$	Instantaneous volume-averaged velocity component
$u_{oi}(\vec{x},t)$	Correlation-estimated instantaneous velocity component
$\Delta u_{oi}(\vec{x},t)=u_{oi}(\vec{x},t)-\tilde{u}_{i}(\vec{x},t)$	Instantaneous velocity component error
$<u^{\prime}_{i}u^{\prime}_{j}>$	Single-point Reynolds stress tensor
$<u^{\prime 2}_{1}>^{vol}$	Volume-averaged turbulence intensity
$v_{i}(\vec{a},t)$	Individual particle Lagrangian velocity
$V(\vec{y})$	Volume of the interrogation volume
$w(\vec{y},t)$	Interrogation volume
$W(\vec{y},\vec{y^{\prime}})$	Triangular window function
$W_{1}(U\delta t)$	Window overlap parameter
$\hat{W}(\vec{k})$	Fourier transform of $W(\vec{y},\vec{y^{\prime}})$
$\vec{x}$	Location vector on physical space
$x_{i}$	Coordinates in physical space, also $(x,y,z)$
$\vec{X}(\vec{a},t)$	Lagrangian displacement field
$\vec{x}_{s}$	Position vector on the detector
$X,Y$	Dimensions of interrogation window (also $\Delta_{1},\Delta_{2}$ )
$\vec{y}$	Position vector of the center of the interrogation volume

$\Delta$	Pixel size
$\Delta_{i}$	Dimension of the interrogation volume along $x_{i}$
$\delta_{i}(\vec{y},\vec{y^{\prime}})=u_{oi}(\vec{y},t)-u_{oi}(\vec{y^{\prime}},t)$	Measured velocity difference between $\vec{y}$ and $\vec{y^{\prime}}$
$\delta_{i}^{noise}(\vec{y},\vec{y^{\prime}})=\Delta u_{oi}(\vec{y},t)-\Delta u_{oi}(\vec{y^{\prime}},t)$	Noise difference between $\vec{y}$ and $\vec{y^{\prime}}$
$\epsilon_{ik}=<\delta_{i}\delta_{k}>$	Second moment of velocity differences
$\zeta$	Noise characteristic constant
$<\eta^{2}>$	Camera noise term
$\mu$	Number of particles per unit volume
$\xi_{ik}$	Noise on $\epsilon_{ik}$
$\sigma$	Noise variance

1 Introduction

A demonstrative application of Particle Image Velocimetry (PIV) was first proposed to the scientific community as ”Speckle Velocimetry” by [meynart83] in 1983. As with all measurement techniques, its theoretical assesment, with the aim of quantifying its accuracy, was developed progressively in parallel to its technical progress. What is specific to PIV, compared to other measurement techniques in fluid mechanics, is the fact that the technology has undergone several revolutions that have affected strongly its characteristics (and this is most probably not finished). Starting from a double-pulsed ruby laser with one set of pulses every 30s, photographic film recording and optical Fourier transform in 1983, we are now using reliable high repetition rate Nd-YAG lasers with low noise high sensitivity CCD or CMOS cameras having millions of pixels, and all the image processing down to the output of velocity vector fields is done in nearly real time even for 3D3C (3 dimensions-3 components) measurements. This can be compared to Hot Wire Anemometry or Laser Doppler Anemometry, for example, for which the configuration was stabilized after about 20 years and the development abandonned progressively by researchers. PIV has been since the beginning (and still is 40 years later) a field of continuous successive technical improvements and impressive creativity of the research community. It is the combination of the availability of always new and better devices, of the progress of the theory and of the inventivity of the scientists which explains mostly this tremendous evolution.

The aim in the present contribution is not to make a historical review of the development of the technique over the years, nor of its theory. The early development of the method is quite well covered by [grant94] and the present status of the theory is well developed in the book of R.J. Adrian and J. Westerweel who are two pioneers of its development [adrianwesterweel2011] and the one of the DLR team [raffel2011] which was also the originator of many significant improvements of the technique.

Beside significant theoretical developments, much understanding has been also gained thanks to the construction of sophisticated synthetic image generators. A good example of such a tool is the generator developed in the frame of the EUROPIV European project [lecordier04]. Its use is well illustrated for example by the contribution of [foucaut04a], but also by the extensive work performed by the scientific community in the frame of the successive PIV challenges organized by the PIVNET European network [stanislas03, stanislas05a, stanislas08]. Finally, several authors have looked at an “a posteriori” estimation of the measurement error, see for example [wieneke15]. Thanks to all these contributions, the PIV technique is quite well understood from the point of view of its theory, and it is possible on this basis to optimize a set-up in order to get the best out of the measurements. Nevertheless, as it is a complex technique involving many different components and physical principles, some routes for further progress are still open such as the scattering characteristics of the particles, the effect of the coherence of the laser light on the particle images shape and the transfer function of the camera, to cite only some of them.

At the early stage of its development PIV could not be considered as an adequate tool to assess turbulence. Limited spatial resolution, limited dynamic range, limited number of samples made it impossible to compete with the Hot Wire Anemometer (HWA) which was the reference tool in turbulence, especially when looking at spectra. But the tremendous improvements encountered by the technique over the course of years have allowed progressively researchers to show measurements of high quality of turbulence statistics with the additional benefit of having access at the same time to the instantaneous turbulence structure [adrian00, carlier05]. Even though the spatial resolution and the number of samples have been greatly improved, the dynamic range of PIV compared to HWA is still a concern; and accurate turbulence statistics measurements require a careful optimization of the recording parameters [foucaut04].

The aim of the present work is to acquire a better understanding of the noise contribution to the statistics of turbulence measured by PIV. For that purpose, focus is not on the technical limitations or imperfections of the devices such as laser beam profile or pixel size, etc. Attention is back to the basics. First the fact that what PIV measures is the instantaneous ‘averaged’ velocity of a set (usually about 10) of particles randomly distributed inside a small interrogation volume. And second the fact that the image is digitized before any processing is applied to it. So in this paper, the PIV tool will be considered perfect: the particles follow exactly the flow, the particle images are all the same and perfectly Gaussian, and the camera is perfectly pixelizing the image. What will be investigated is how far the result provided by this perfect tool is from the instantaneously averaged velocity of the fluid inside an interrogation volume where the local velocity is not uniform. To do so, we examine PIV theoretically using the methodologies developed earlier by [GeorgeLumley1973], [george88], [Buchhaveetal1979], and most recently [Velteetal2014] to study Laser Doppler Anemometry.

As the authors have no possibility to perform an experiment themselves, an extensive use is made of the results of a study of the PIV noise by [foucaut04]. The main interest of this experiment is that measurements were performed both in a fluid at rest and in a turbulent flow with different recording parameters. The analysis of these data allowed the authors to derive an empirical model for the noise and the PIV spectrum. As will be seen, the present theory not only validates this model, but provides also explicit theoretical expressions for the empirical parameters.

2 Determining the PIV “output”

2.1 A simplified PIV model

This section follows almost exactly the formulation used in many papers by Adrian and Westerweel [westerweel13], and most notably their book [adrianwesterweel2011]. The primary difference is the use of generalized functions instead of finite (but time-dependent) sums. It is slightly more general, at least at the beginning, since it uses Lagrangian coordinates in which the particle initial locations can be assumed to be statistically independent of their motion, an assumption generally not valid once turbulence moves them around.

In our model we imagine that each particle creates an image which moves across a detector as the particle moves. We imagine the system to have been perfectly aligned and calibrated, and all the particles are assumed to create scaled versions of exactly the same image in the detector-plane, say $b_{cam}(\vec{x_{s}},t)$ , where $\vec{x_{s}}=(x_{s},y_{s})$ is the position vector on the detector. We could as well include the varying amplitude because of their size and position within the light beam, but for now we will not.

By appropriate coordinate transformation, we can link each of these images to the particle coordinate in the flow, say $\vec{x}=(x,y,z)$ . The origin of both coordinate systems can be chosen arbitrarily, so let’s choose the origins of $x$ and $x_{s}$ to coincide, and $y$ and $y_{s}$ to coincide. We choose the origin of $z$ to be in the center of the illuminating laser sheet. Since there is assumed to be a one-to-one mapping between the position of the particle within the beam and its position on the camera backplane, we can choose instead to work in the flow coordinate system; i.e., we can write the image produced by each particle as