Corrections for one- and two-point statistics measured with coarse-resolution particle image velocimetry

Abstract

A theoretical model to determine the effect of the size of the interrogation window in particle image velocimetry measurements of turbulent flows is presented. The error introduced by the window size in two-point velocity statistics, including velocity autocovariance and structure functions, is derived for flows that are homogeneous within a 2D plane or 3D volume. This error model is more general than those previously discussed in the literature and provides a more direct method of correcting biases in experimental data. Within this model framework, simple polynomial approximations are proposed to provide a quick estimation of the effect of the averaging on these statistics. The error model and its polynomial approximation are validated using statistics of homogeneous isotropic turbulence obtained in a physical experiment and in a direct numerical simulation. The results demonstrate that the present formulation is able to correctly estimate the turbulence statistics, even in the case of strong smoothing due to a large interrogation window. We discuss how to use these results to correct experimental data and to aid the comparison of numerical results with laboratory data.

Relationship between interrogation window and measured velocity

To quantify the effect of the interrogation window on the measured velocity, let us consider two PIV images taken at two different instants $t_0$ and $t_1=t_0+\Delta t$. It is expected that these images will be black (zero light intensity) almost everywhere, with the exception of some points where the laser light reflected by the particles is detected. We assume now a square interrogation area, $I_0$, of size $L$ where $N$ particles are located in the first image, and another interrogation area in the second image, $I_1$, of the same size as $I_0$ but translated with a convection velocity $\varvec{V}_m$, for the moment undefined. The location of the illuminated points can be labeled as $\varvec{x}_{0,i}$ in $I_0$ and $\varvec{x}_{1,i}=\varvec{x}_{0,i}+\left( \varvec{V}_i-\varvec{V}_m\right) \Delta t$ in $I_1$ with $i\in \lbrace 1,2,...,N\rbrace$, where $\varvec{V}_i$ denotes the average velocity of the $i^{\mathrm{th}}$-particle between $t_0$ and $t_1$. It is assumed that there are only a negligible number of particles leaving the domain determined by the interrogation area, so that the present analysis has general validity. The light intensity distribution over the two interrogations areas can be expressed as

$$\begin{aligned} I_0\left( \varvec{x}\right) =\sum _{i=1}^N\rho \left( \varvec{x}-\varvec{x}_{0,i}\right) \quad \hbox {and}\quad I_1\left( \varvec{x}\right) =\sum _{i=1}^N\rho \left[ \varvec{x}-\varvec{x}_{0,i}-\left( \varvec{V}_i-\varvec{V}_m\right) \Delta t\right] , \end{aligned}$$

(22)

where $\rho \left( \varvec{x}\right)$ is a function that represents the light intensity around a particle located at the origin. For the sake of simplicity, it will be assumed to be a rapidly decaying Gaussian.

The cross-correlation operator between the two images can now be introduced as

$$\begin{aligned} R\left( \varvec{\tau }\right) =\int \limits _DI_0\left( \varvec{x}\right) I_1\left( \varvec{x}+\varvec{\tau }\right) \hbox {d}\varvec{x}, \end{aligned}$$

(23)

where $D$ is a square domain of size $L$ that includes the interrogation area. The cross-correlation, together with Eq. (22), becomes

$$\begin{aligned} R\left( \varvec{\tau }\right) =\sum _{i=1}^N\sum _{j=1}^N\int \limits _D\rho \left( \varvec{x}-\varvec{x}_{0,i}\right) \rho \left[ \varvec{x}+\varvec{\tau }-\varvec{x}_{0,j}-\left( \varvec{V}_j-\varvec{V}_m\right) \Delta t\right] \hbox {d}\varvec{x}. \end{aligned}$$

(24)

The maximum of the cross-correlation function identifies the optimal interrogation window displacement, $\varvec{\tau }$, that ensures the highest correlation. Therefore, $\varvec{V}_m$ can be seen as the convective velocity maximizing $R\left( {0}\right)$. The maximum of the cross-correlation is readily obtained by imposing that the gradient must be zero at $\varvec{\tau }={0}$ so that

$$\begin{aligned} \frac{\partial R}{\partial x_k}=\sum _{i=1}^N\sum _{j=1}^N\int \limits _D\rho \left( \varvec{x}-\varvec{x}_{0,i}\right) \frac{\partial \rho }{\partial x_k}\left[ \varvec{x}-\varvec{x}_{0,j}-\left( \varvec{V}_j-\varvec{V}_m\right) \Delta t\right] \hbox {d}\varvec{x}=0. \end{aligned}$$

(25)

To proceed further, it is possible to assume that the term $\left( \varvec{V}_j-\varvec{V}_m\right) \Delta t$ in Eq. (25) is small, so that a simple Taylor expansion can be used

$$\begin{aligned} \frac{\partial R}{\partial x_k}&= \sum _{i=1}^N\sum _{j=1}^N\int \limits _D\rho \left( \varvec{x}-\varvec{x}_{0,i}\right) \frac{\partial \rho }{\partial x_k}\left( \varvec{x}-\varvec{x}_{0,j}\right) \hbox {d}\varvec{x}\nonumber \\&\quad -\left( \varvec{V}_j-\varvec{V}_m\right) _n\Delta t\int \limits _D\rho \left( \varvec{x}-\varvec{x}_{0,i}\right) \frac{\partial ^2\rho }{\partial x_k\partial x_n}\left( \varvec{x}-\varvec{x}_{0,j}\right) \hbox {d}\varvec{x}=0. \end{aligned}$$

(26)

The first integral is zero for $i=j$ since the function $\rho$ is assumed to be isotropic, while for $i\ne j$ is approximately zero since it is assumed that no particles lie near the boundaries. The second integral is nonzero if $i=j$ and becomes negligible otherwise (since $\rho$ is rapidly decaying). Therefore, Eq. (26) can be approximated by considering only the second integral when $i=j$, and by noting that $\rho$ is assumed to be the same for all particles

$$\begin{aligned} \int \limits _D\rho \left( \varvec{x}-\varvec{x}_{0,1}\right) \frac{\partial ^2\rho }{\partial x_k\partial x_n}\left( \varvec{x}-\varvec{x}_{0,1}\right) \hbox {d}\varvec{x}\sum _{i=1}^N\left( \varvec{V}_i-\varvec{V}_m\right) _n=0\quad\mathrm{namely}\quad \varvec{V}_m=\frac{1}{N}\sum _{i=1}^N\varvec{V}_i. \end{aligned}$$

(27)

The measured velocity is therefore the arithmetic mean of the average velocity (within $\Delta t$) of the $N$ particles inside the interrogation area $D$.

Since the particles are assumed to be embedded in a velocity field $\varvec{V}\left( \varvec{x},t\right)$ that is homogeneous in the image plane, they are uniformly distributed in space. The expected velocity of the generic $i{\mathrm{th}}$-particle inside $D$ is statistically equal to the average flow velocity, assuming ideal particles with no inertia, therefore

$$\begin{aligned} \varvec{V}_i=\frac{1}{L^2}\int \limits _D\varvec{V}\left( \varvec{x},t\right) \hbox {d}\varvec{x}, \end{aligned}$$

(28)

and the measured velocity can be expressed using Eq. (27) as

$$\begin{aligned} \varvec{V}_m=\frac{1}{L^2}\int \limits _D\varvec{V}\left( \varvec{x},t\right) \hbox {d}\varvec{x}. \end{aligned}$$

(29)

In conclusion, the measured velocity is approximately equal to the integral average of the velocity field inside the interrogation area.