PIV uncertainty propagation

Consider the generic velocity component u. Based on equations (7) and (11), the uncertainty of the mean velocity $\bar{u}$ is:

$$\begin{eqnarray}&&{{U}_{{\bar{u}}}}=\frac{{{\sigma}_{u}}}{\sqrt{N}}\end{eqnarray} \tag{ 15 }$$

Analogous equations are obtained for the v and w velocity components. In equation (15), systematic uncertainties due to spatial modulation errors or peak locking are not taken into account. The standard deviation σ_u contains both the true velocity fluctuations (σ_{u, fluct}) and the measurement errors (σ_u,err):

$$\begin{eqnarray}&&\sigma _{u}^{2}=\sigma _{u,\,\text{fluct}}^{2}+\sigma _{u,\,\text{err}}^{2}\approx \sigma _{u,\,\text{fluct}}^{2}+\overline{U_{u}^{2}}\end{eqnarray} \tag{ 16 }$$

where U_u is the uncertainty of the instantaneous velocity component and $\overline{U_{u}^{2}}$ is the mean-square of U_u. The right-hand-side of equation (16) is obtained by considering that the error variance $\sigma _{u,\,\text{err}}^{2}$ is approximately equal to the uncertainty mean-square $\overline{U_{u}^{2}}$ for accurate uncertainty quantification methods (see appendix of Sciacchitano et al 2015).

When the samples are not independent, the parameter N of equation (15) must be substituted with the effective number of independent samples N_eff, as discussed in section 2.2.3.

The Reynolds stress plays a crucial role in turbulent flows because it represents the rate of mean momentum transfer by turbulent fluctuations. In this section, the expression of the uncertainty is derived for the Reynolds normal stress and for the Reynolds shear stress.

Reynolds normal stress.

The Reynolds normal stress for the x-velocity component u is defined as the variance of u:

$$\begin{eqnarray}&&{{R}_{uu}}=\overline{{{u}^{\prime 2}}}=\sigma _{u}^{2}=\frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,{{\left({{u}_{i}}-\bar{u}\right)}^{2}}\end{eqnarray} \tag{ 17 }$$

where $u'$ is the fluctuating part of u: ${{u}^{\prime}}=u-\bar{u}$. Due to its definition, the uncertainty of R_uu is computed with equation (13):

$$\begin{eqnarray}&&{{U}_{{{R}_{uu}}}}=\sigma _{u}^{2}\sqrt{\frac{2}{N-1}}\cong \sigma _{u}^{2}\sqrt{\frac{2}{N}}={{R}_{uu}}\sqrt{\frac{2}{N}}\end{eqnarray} \tag{ 18 }$$

It is assumed that the samples are statistically independent. If not, N must again be substituted with the effective number of independent samples N_eff (section 2.2.3). As discussed in section 2.2.1, σ_u contains both the effects of true velocity fluctuations and spurious fluctuations due to noise. The latter yield an over-estimate for R_uu with respect to the true value R_{uu, true}:

$$\begin{eqnarray}&&{{R}_{uu}}={{R}_{uu,\text{true}}}+\sigma _{u,\text{err}}^{2}={{R}_{uu,\text{true}}}+\overline{U_{u}^{2}}\end{eqnarray} \tag{ 19 }$$

When the uncertainty of the measured velocity is known, a corrected (more accurate) estimate of R_uu can be retrieved by subtracting the spurious fluctuations mean square $\overline{U_{u}^{2}}$ from the measured Reynolds stress:

$$\begin{eqnarray}&&{{R}_{uu,\text{corr}}}={{R}_{uu}}-\overline{U_{u}^{2}}\end{eqnarray} \tag{ 20 }$$

Thus, according to equation (6), the uncertainty of the corrected normal Reynolds stress estimate R_{uu, corr}, indicated with ${{U}_{{{R}_{uu,\text{corr}}}}}$, is given by:

$$\begin{eqnarray}&&{{U}_{{{R}_{uu,\text{corr}}}}}=\sqrt{U_{{{R}_{uu}}}^{2}+U_{\overline{U_{u}^{2}}}^{2}}\end{eqnarray} \tag{ 21 }$$

The latter is composed by two components: (a) the uncertainty of the measured Reynolds stress, which is given by equation (18); (b) the uncertainty of the spurious fluctuations mean square $\overline{U_{u}^{2}}$. Notice that $U_{u}^{2}$ can only assume positive values, therefore its distribution is better approximated by a log-normal distribution rather than by a Gaussian distribution. At least when approximating the distribution with a Gaussian distribution with positive mean, an analytical expression of the uncertainty of $\overline{U_{u}^{2}}$ can be derived using equation (6):

$$\begin{eqnarray}&&{{U}_{\overline{U_{u}^{2}}}}=\frac{2}{\sqrt{N}}\,\sigma _{{{U}_{u}}}^{{}}\overline{{{U}_{u}}}\centerdot \sqrt{1+\frac{\sigma _{{{U}_{u}}}^{2}}{2{{\overline{{{U}_{u}}}}^{2}}}}\end{eqnarray} \tag{ 22 }$$

The accuracy of equation (22) is assessed in section 3.1. Combining both equations (18) and (22), the uncertainty of R_{uu, corr} is:

$$\begin{eqnarray}&&{{U}_{{{R}_{uu,\text{corr}}}}}=\sqrt{R_{uu}^{2}+{{\left(\sqrt{2}\,{{\sigma}_{{{U}_{u}}}}\overline{{{U}_{u}}}\centerdot \sqrt{1+\frac{\sigma _{{{U}_{u}}}^{2}}{2{{\overline{{{U}_{u}}}}^{2}}}}\right)}^{2}}}\centerdot \sqrt{\frac{2}{N}}\end{eqnarray} \tag{ 23 }$$

In many applications, the measurement error is small with respect to the actual velocity fluctuations, therefore the term within brackets is negligible and equation (23) reduces to:

$$\begin{eqnarray}&&{{U}_{{{R}_{uu,\text{corr}}}}}={{R}_{uu}}\centerdot \sqrt{\frac{2}{N}}\end{eqnarray} \tag{ 24 }$$

In practice, the uncertainty of the Reynolds normal stress according to (23) and (24) is often strongly underestimated for two reasons. First, the subtraction of equation (20) is subject to the accuracy of the uncertainty quantification method itself. As shown by Sciacchitano et al (2015), the uncertainty estimations of state-of-the-art UQ methods may deviate from the true errors by as much as a factor two for different flow and imaging conditions. Secondly, the finite spatial resolution of the PIV processing algorithm does not allow resolving fluctuations of length scale smaller than about the interrogation window. This may lead to a substantial underestimation of R_uu depending on Reynolds number, turbulent level and imaging magnification.

It is important to remark here that the computation of the uncertainty of R_uu according to equation (18) does not require the knowledge of the uncertainty of the instantaneous velocity. On the other hand, in order to compute the corrected value R_{uu, corr}, the uncertainty of the instantaneous velocity must be known.

Turbulent kinetic energy.

The turbulent kinetic energy TKE is defined as half of the sum of the Reynolds normal stresses:

$$\begin{eqnarray}&&\text{TKE}=\frac{1}{2}\overline{u{{_{i}^{\prime}}_{{}}}u_{i}^{\prime}}=\frac{1}{2}\left({{R}_{uu}}+{{R}_{vv}}+{{R}_{ww}}\right)\end{eqnarray} \tag{ 25 }$$

Based on the error propagation formula (6), the uncertainty of the TKE is equal to:

$$\begin{eqnarray}&&{{U}_{\text{TKE}}}=\frac{1}{2}\sqrt{U_{{{R}_{uu}}}^{2}+U_{{{R}_{vv}}}^{2}+U_{{{R}_{ww}}}^{2}}\end{eqnarray} \tag{ 26 }$$

Assuming N $\gg$ 1 and that the instantaneous measurement uncertainty is negligible with respect to the velocity fluctuations, the result of equation (24) can be used and the expression of U_TKE reduces to:

$$\begin{eqnarray}&&{{U}_{\text{TKE}}}=\sqrt{R_{uu}^{2}+R_{vv}^{2}+R_{ww}^{2}}\centerdot \sqrt{\frac{1}{2N}}\end{eqnarray} \tag{ 27 }$$

When R_ww is unknown (e.g. in planar PIV, which only provides two velocity components), its value can be estimated as ${{R}_{ww}}=\left({{R}_{uu}}+{{R}_{vv}}\right)/2$ under the assumption of isotropic turbulence.

Reynolds shear stress.

The Reynolds shear stress R_uv is defined as the covariance of the u and v velocity components:

$$\begin{eqnarray}&&{{R}_{uv}}=\overline{{{u}^{\prime}}{{v}^{\prime}}}=\frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,u{{_{i}^{\prime}}_{{}}}v_{i}^{\prime}=\frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,\left({{u}_{i}}-\bar{u}\right)\left({{v}_{i}}-\bar{v}\right)\\ &&={{\rho}_{uv}}{{\sigma}_{u}}{{\sigma}_{v}}\end{eqnarray} \tag{ 28 }$$

The quantity ρ_uv is the cross-correlation coefficient between the velocity components u and v. Assuming that the velocity fluctuations are affected by error δu and δv, respectively, and that the error of the time-averaged velocity is negligible, equation (28) becomes:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{R}_{uv}}= & \frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,\left(u_{i,\text{true}}^{\prime}+\delta u\right)\left(v_{i,\text{true}}^{\prime}+\delta v\right) \\ = & \frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,\left(u_{i,\text{true}}^{\prime}v_{i,\text{true}}^{\prime}+u_{i,\text{true}}^{\prime}\delta v+v_{i,\text{true}}^{\prime}\delta u+\delta u\delta v\right) \\ = & \frac{1}{N-1}\underset{i=1}{\overset{N}{\sum}}\,\left(u_{i,\text{true}}^{\prime}v_{i,\text{true}}^{\prime}+\delta u\delta v\right) \\ = & {{R}_{uv,\text{true}}}+{{\rho}_{\delta u\delta v}}{{\sigma}_{\delta u}}{{\sigma}_{\delta v}}\cong {{R}_{uv,\text{true}}}+{{\rho}_{\delta u\delta v}}\sqrt{\overline{U_{u}^{2}}\,\overline{U_{v}^{2}}} \end{array}\end{eqnarray} \tag{ 29 }$$

In equation (29), ρ_δuδv is the cross-correlation coefficient between the errors of the two velocity components. The true velocity fluctuations are assumed to be independent of the measurement errors, thus cancelling the cross-terms ${\sum}_{i=1}^{N}\left(u_{i,\text{true}}^{\prime}\delta v\right)$ and ${\sum}_{i=1}^{N}\left(v_{i,\text{true}}^{\prime}\delta u\right)$. As a consequence, the Reynolds shear stress R_uv exhibits a systematic error (equal to ${{\rho}_{\delta u\delta v}}\sqrt{\overline{U_{u}^{2}}\,\overline{U_{v}^{2}}}$) only if δu and δv are correlated (ρ_δuδv  ≠  0); however, this is typically not the case for planar 2 C-PIV. Conversely, for stereo-PIV there may be non-zero inter- component correlations dependent on the experimental setup of the two cameras relative to the x- and y-axis. The uncertainty of R_uv is obtained by applying the covariance uncertainty equation (14):

$$\begin{eqnarray}&&{{U}_{{{R}_{uv}}}}={{\sigma}_{u}}{{\sigma}_{v}}\centerdot \sqrt{\frac{1+\rho _{uv}^{2}}{N-1}}\end{eqnarray} \tag{ 30 }$$

The uncertainty of the Reynolds shear stress has a minimum value of ${{\sigma}_{u}}{{\sigma}_{v}}/\sqrt{N-1}$ when u and v are uncorrelated and increases with higher correlation between the two velocity components.

Consider a generic statistical quantity, as the mean $\bar{x}$. In this section we will show that if the N samples from which $\bar{x}$ is computed are not independent, a larger uncertainty of $\bar{x}$ is expected. In fact, from equation (6) it is obtained:

$$\begin{eqnarray}&&U_{{\bar{x}}}^{2}=\underset{i=1}{\overset{N}{\sum}}\,\underset{j=1}{\overset{N}{\sum}}\,\frac{1}{{{N}^{2}}}\rho \left({{x}_{i}},{{x}_{j}}\right)\sigma _{x}^{2}\end{eqnarray} \tag{ 31 }$$

having assumed a constant underlying fluctuation distribution ${{\sigma}_{{{x}_{i}}}}={{\sigma}_{{{x}_{j}}}}={{\sigma}_{x}}$. The auto-correlation coefficient ρ(x_i, x_j) can be written as:

$$\begin{eqnarray}&&\rho \left({{x}_{i}},{{x}_{j}}\right)=\rho \left({{x}_{i}},{{x}_{i+n}}\right)=\rho (n\Delta t)\end{eqnarray} \tag{ 32 }$$

with Δt the inverse of the sampling frequency. The auto-correlation coefficient ρ is a function of the time separation nΔt between samples x_i and x_j  =  x_i+n. As a result, equation (31) can be written as:

$$\begin{eqnarray}&&U_{{\bar{x}}}^{2}=\frac{\sigma _{x}^{2}}{{{N}^{2}}}\underset{i=1}{\overset{N}{\sum}}\,\underset{n=1-i}{\overset{N-i}{\sum}}\,\rho \left({{x}_{i}},{{x}_{i+n}}\right)=\frac{\sigma _{x}^{2}}{{{N}^{2}}}\underset{i=1}{\overset{N}{\sum}}\,\underset{n=1-i}{\overset{N-i}{\sum}}\,\rho (n\Delta t)\end{eqnarray} \tag{ 33 }$$

The quantity ρ(nΔt) is equal to one for n  =  0 and decays to zero for increasing n. Furthermore, ρ(nΔt) is an even function: ρ(nΔt)  =  ρ(–nΔt). Assuming $N\to \infty$ and neglecting the edge effects in the summation, equation (33) becomes:

$$\begin{eqnarray}&&U_{{\bar{x}}}^{2}=\frac{\sigma _{x}^{2}}{{{N}^{2}}}\underset{i=1}{\overset{N}{\sum}}\,\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)=\frac{\sigma _{x}^{2}}{{{N}^{2}}}N\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)\\ &&=\sigma _{x}^{2}\frac{\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)}{N}\,\end{eqnarray} \tag{ 34 }$$

Defining the effective number of independent samples as:

$$\begin{eqnarray}&&{{N}_{\text{eff}}}=\frac{N}{\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)}\,\end{eqnarray} \tag{ 35 }$$

leads to:

$$\begin{eqnarray}&&U_{{\bar{x}}}^{2}=\frac{\sigma _{x}^{2}}{{{N}_{\text{eff}}}}\text{or} ~{{U}_{{\bar{x}}}}=\frac{{{\sigma}_{x}}}{\sqrt{{{N}_{\text{eff}}}}}\end{eqnarray} \tag{ 36 }$$

Typically, the summation of equation (35) is stopped when the correlation value reaches zero for the first time. Notice that when the samples are uncorrelated, then ρ(nΔt) is 1 for n  =  0 and zero otherwise, so in this case N_eff  =  N. Conversely, when the samples are correlated then ${\sum}_{n=-\infty}^{+\infty}\rho (n\Delta t)>1$; therefore N_eff is smaller than N, thus the uncertainty of the mean value is larger.

The integral time scale T_int is defined as the integral of the auto-correlation function ρ(t) of the time series x(t) (George et al 1978):

$$\begin{eqnarray}&&{{T}_{\operatorname{int}}}={\int}_{0}^{\infty}\rho (t)\text{d}t\end{eqnarray} \tag{ 37 }$$

T_int is a measure of the time interval over which x(t) is dependent on itself. For time intervals large compared to T_int, x(t) becomes statistically independent of itself. Then, the effective number of independent samples can be written as a function of the observation time T and the integral time scale T_int:

$$\begin{eqnarray}&&{{N}_{\text{eff}}}=\frac{N}{\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)}=\frac{N\centerdot \Delta t}{\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (n\Delta t)\centerdot \Delta t}\\ &&\approx \frac{T}{{\int}_{-\infty}^{+\infty}\rho \left(\text{d}t\right)\text{d}t}=\frac{T}{2{\int}_{0}^{+\infty}\rho \left(\text{d}t\right)\text{d}t}=\frac{T}{2{{T}_{\operatorname{int}}}}\end{eqnarray} \tag{ 38 }$$

The relevance of equations (36) and (38) for experimental measurements in turbulent flows is discussed by Tennekes and Lumley (1972) among others. The equations illustrate the fact that, when Δt  <  T_int and the total observation time T is fixed, increasing the sampling frequency and therefore the number of samples does not improve the accuracy of the derived statistical quantities (Taylor 1997), because the effective number of independent samples stays constant. Instead, it is advantageous to limit the sampling frequency to 1/(2T_int) and increase the recording time T.

Let us consider a planar-PIV measurement where the velocity components (u, v) are measured in a 2D domain. The out-of-plane vorticity component is defined as:

$$\begin{eqnarray}&&{{\omega}_{z}}=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\end{eqnarray} \tag{ 39 }$$

For sake of brevity, we will drop the subscript z in the reminder and we will indicate the out-of-pane vorticity component simply with ω. The velocity components u and v are discrete functions, defined at grid points with uniform spacing d (both in x- and y-direction). As an example, the vorticity can be computed by the central-difference scheme by:

$$\begin{eqnarray}&&\omega (x,y)=\frac{1}{2d}\left[v(x+d,y)-v(x-d,y)-u(x,y+d)\right. \\ &&\left. +u(x,y-d)\right]\end{eqnarray} \tag{ 40 }$$

Other methods using larger kernel sizes are available at the expense of lower spatial resolution of the vorticity field. Using the error propagation formula (6), the uncertainty of the vorticity at grid point (x, y) is (apart from truncation errors):

$$\begin{eqnarray}\begin{array}{*{35}{l}} U_{\omega}^{2}= & {{\left(\frac{1}{2d}\right)}^{2}}\left[U_{v}^{2}+U_{v}^{2}+U_{u}^{2}+U_{u}^{2}-2\rho (2d)U_{v}^{2}-2\rho (2d)U_{u}^{2}\right] \\ = & 2{{\left(\frac{1}{2d}\right)}^{2}}\left[1-\rho (2d)\right]\left(U_{u}^{2}+U_{v}^{2}\right) \end{array}\end{eqnarray} \tag{ 41 }$$

where the following assumptions have been made:

• i.  
  The errors of u and v at the same or neighboring spatial locations are uncorrelated (2C-PIV).
• ii.  
  The errors of u(x, y  +  d) and u(x, y  −  d) are spatially correlated (and similarly the errors of v(x  +  d, y) and v(x  −  d, y)). The corresponding cross-correlation coefficient, indicated with ρ(2d), is assumed to be the same for the two velocity components. It represents the normalized cross-correlation of the measurement error at two grid points at spatial separation 2d.
• iii.  
  The uncertainty of u(x, y  +  d) is assumed to be equal to the uncertainty of u(x, y  −  d) and is indicated with U_u. Likewise, the uncertainty of v(x  +  d, y) is assumed to be equal to the uncertainty of v(x  −  d, y) and is indicated with U_v. In practice, an appropriate local average of uncertainties can be taken.

If we further assume that the two velocity components have the same uncertainty (U_u  =  U_v  =  U), the expression of the uncertainty of the vorticity simplifies to:

$$\begin{eqnarray}&&{{U}_{\omega}}=\frac{U}{d}\sqrt{1-\rho (2d)}\end{eqnarray} \tag{ 42 }$$

Equation (42) shows the proportionality between the uncertainties of vorticity and velocity. The grid spacing d has a twofold effect on U_ω: on the one hand, U_ω is inversely proportional to d, which would cause a reduction of U_ω when d is increased. On the other hand, increasing d yields a reduction of the spatial cross-correlation coefficient and in turn an increase of the square-root term. When the interrogation window overlap is increased, d tends to zero faster than $\sqrt{1-\rho (2d)}$: as a consequence, the uncertainty of the vorticity increases (see figure 1). However, two things should be kept in mind: (a) equation (42) accounts only for the random errors of the vorticity and not for the truncation errors, which are systematic and decrease when increasing the interrogation window overlap; (b) the uncertainty of the vorticity can be reduced by computing the spatial derivatives using a larger spacing in the finite differences (e.g. using $\left[v(x+2d,y)-v(x-2d,y)\right]/4d$ instead of $\left[v(x+d,y)-v(x-d,y)\right]/2d$). For noisy data, Vollmers (2001) reports that lower uncertainty can be achieved by computing the vorticity from the flow circulation, rather than via equations (39) and (40). Linear error propagation can be used to evaluate the uncertainty of the vorticity calculated with advanced algorithms; the determination and analysis of that uncertainty goes beyond the scope of the present paper.

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It can be shown that equation (42) also corresponds to the uncertainty of the 2D divergence of the velocity. Conversely, for 3D divergence in tomographic PIV, the following expression of the uncertainty is derived:

$$\begin{eqnarray}&&{{U}_{\text{div}}}=\frac{U}{d}\sqrt{\frac{3}{2}\left[1-\rho (2d)\right]}\end{eqnarray} \tag{ 43 }$$

The above derivations can be modified accordingly when the central-difference scheme is replaced by more elaborate functions, e.g. fitting flow derivatives by a Levenberg–Marquardt algorithm on a 3  ×  3 or 5  ×  5 vector kernel size. For stereo-PIV with non-zero correlations between u and v, additional terms must be taken into account in the above equations. Stereo-PIV uncertainty quantification including assessment of calibration errors will be subject of future work.

When a velocity component is spatially averaged over a profile, region or volume, the uncertainty of the average could be computed either by equation (11) using the fluctuations of the velocity vectors, or, alternatively, by considering the mean as a simple function and propagating the individual velocity uncertainties according to equation (6). Usually the second option is preferred, since the mean should be considered here as an instantaneous quantity and not as a statistically converged value. Most often, the underlying mean and standard deviation will be anyway different at different spatial locations. Only in the case of averaging over isotropic homogeneous turbulence with sufficient data points one could try to measure turbulent statistical values; even in this case, it would be more accurate to record a large number of images over time for unbiased statistics.

The derivation of the uncertainty of the mean is done in the same way as in equations (31)–(36), replacing standard deviations with uncertainties, and replacing temporal correlation of velocity components with the spatial correlation of the velocity errors, which are closely related to the spatial resolution of the PIV processing scheme.

Consider the 1D-case with N values of the u velocity component averaged along a profile in x-direction:

$$\begin{eqnarray}&&\bar{u}=\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\,{{u}_{i}}\end{eqnarray} \tag{ 44 }$$

According to equation (6), the uncertainty of the mean is:

$$\begin{eqnarray}&&U_{{\bar{u}}}^{2}=\underset{i=1}{\overset{N}{\sum}}\,\underset{j=1}{\overset{N}{\sum}}\,\frac{1}{{{N}^{2}}}\rho \left(\delta {{u}_{i}},\delta {{u}_{j}}\right){{U}_{{{u}_{i}}}}{{U}_{{{u}_{j}}}}\approx \underset{i=1}{\overset{N}{\sum}}\,\underset{j=1}{\overset{N}{\sum}}\,\frac{1}{{{N}^{2}}}\rho \left(\delta {{u}_{i}},\delta {{u}_{j}}\right)\overline{U_{u}^{2}}\end{eqnarray} \tag{ 45 }$$

where, for simplification, the product of individual uncertainties ${{U}_{{{u}_{i}}}}{{U}_{{{u}_{j}}}}$ is substituted by the mean square uncertainty $\overline{U_{u}^{2}}$. The spatial auto-correlation coefficients ρ(δu_i, δu_j) can be written as a function of the vector grid spacing d:

$$\begin{eqnarray}&&\rho \left(\delta {{u}_{i}},\delta {{u}_{j}}\right)=\rho \,\left(|\,j-i|d\right)=\rho (nd)\end{eqnarray} \tag{ 46 }$$

where n is the number of grid points between locations i and j. Neglecting edge effects, i.e. requiring large N, equation (45) leads to:

$$\begin{eqnarray}&&U_{{\bar{u}}}^{2}=\frac{\overline{U_{u}^{2}}}{{{N}^{2}}}\underset{i=1}{\overset{N}{\sum}}\,\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (nd)=\frac{\overline{U_{u}^{2}}}{{{N}^{2}}}N\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (nd)\\ &&=\frac{\overline{U_{u}^{2}}}{N}\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (nd)\end{eqnarray} \tag{ 47 }$$

Again, an effective number of independent samples can be defined as:

$$\begin{eqnarray}&&{{N}_{\text{eff}}}=\frac{N}{\underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (nd)}\end{eqnarray} \tag{ 48 }$$

thus:

$$\begin{eqnarray}&&{{U}_{{\bar{u}}}}=\frac{U_{u}^{\text{rms}}}{\sqrt{{{N}_{\text{eff}}}}}\end{eqnarray} \tag{ 49 }$$

having defined the root-mean-square averaged uncertainty $U_{u}^{\text{rms}}=\sqrt{\overline{U_{u}^{2}}}$.

The integral of the auto-correlation coefficients can be defined as the spatial resolution L_sr of the PIV algorithm, which in pixel units is:

$$\begin{eqnarray}&&{{L}_{\text{sr}}}={\int}_{-\infty}^{+\infty}\rho (x)\text{d}x\end{eqnarray} \tag{ 50 }$$

The spatial resolution can also be written relative to the vector spacing d:

$$\begin{eqnarray}&&L_{\text{sr}}^{\ast}=\frac{{\int}_{-\infty}^{+\infty}\rho (x)\text{d}x}{d}\approx \underset{n=-\infty}{\overset{+\infty}{\sum}}\,\rho (nd)\end{eqnarray} \tag{ 51 }$$

In the 2D-case, when the average is conducted over a region of N_x  ×  N_y vectors, equation (48) becomes:

$$\begin{eqnarray}&&{{N}_{\text{eff}}}=\frac{{{N}_{x}}{{N}_{y}}}{L_{\text{sr}}^{\ast \,2}}\end{eqnarray} \tag{ 52 }$$

with $L_{\text{sr}}^{\ast \,}$ again in units of vector spacing and assuming the same spatial resolution in x and y. This is, for example, not the case for advanced locally adaptive PIV schemes with elongated windows e.g. adjusting to boundaries. For a single-pass PIV processing scheme with a square interrogation window of n_pix  ×  n_pix pixel, the correlation function ρ(x) is the triangle function $\left(1-\frac{|x|}{{{n}_{\text{pix}}}}\right)$ for |x|  ⩽  n_pix, and 0 otherwise. Hence, the spatial resolution is simply L_sr  =  n_pix. For a Gaussian weighted interrogation window with a standard deviation of σ, it can be shown that the spatial resolution is equal to ${{L}_{\text{sr}}}=\sqrt{4\pi}\sigma$. For state-of-the-art PIV algorithms using multi-pass window deformation (like DaVis 8), it has been found that the correlation function—when approximating the PIV algorithm as a linear spatial filter function—is Gaussian with some Mexican hat contribution leading to slight overshooting for steep velocity step functions as observed by Elsinga and Westerweel (2011). A detailed analysis is beyond the scope of this work.

In practice, the correlation coefficients and spatial resolution need to be specified for a particular set of PIV processing parameters. When the averaging process is conducted with a small number of vectors over a region comparable to the spatial resolution, the simplifying assumptions that led to equation (49) are not valid anymore. In this case, the uncertainty of the spatial mean must be computed via equation (6), where all individual correlation coefficients must be taken into account.

The errors of neighboring vectors are spatially correlated due to the interrogation window overlap. To investigate the spatial correlation of the error, a Monte Carlo simulation is conducted considering a null displacement field. The images have a resolution of 5000  ×  400 pixels, with a seeding concentration of 0.05 ppp. The particle images have a Gaussian intensity profile with peak intensity of 1024 counts; their diameter is set to 2 pixels. Noise is added to the recordings (white background noise with 5 counts standard deviation and photon shot noise, assuming a conversion factor of 4 electrons per count) to cause errors in the measured velocity. The images are processed with the commercial software DaVis 8.2 from LaVision. The auto-correlation function ρ of the measurement error is computed to investigate the spatial correlation among neighboring vectors. The results of figure 2, which refer to the case of Gaussian-weighted interrogation window size of 32  ×  32 pixels with 75% overlap, show that a significant correlation is present up to sample spacing of 3d. Notice that in this case ρ(2d)  ≅  0.45; hence, the spatial correlation of the error is relevant and cannot be neglected for the computation of the uncertainty of the vorticity via equation (42). Note that the above mentioned mixture of Gaussian and Mexican hat filter function of PIV leads here to the slight undershooting of the correlation values below zero.

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To assess the uncertainty of the vorticity, the rectangular jet data are used. The velocity time series is extracted at a point P  =  (398, 246) as shown in figure 10. Figure 12 shows a portion of the time series for a time interval of 10 ms. The comparison between MS and HDR data on the entire time series yields the error for the MS reported in table 4. It is noticed that: (a) the two error components δu and δv have comparable magnitude; (b) the random component of the error (error standard deviation) is significantly larger than the mean bias component.

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The vorticity is computed with the central-difference scheme of equation (40), with grid spacing d  =  4 px. The vorticity time series for the first 10 ms is shown in figure 13. Both HDR and MS yield the same peak vorticity (ω_max  =  – 0.15 px/px at t  =  2.2 ms), which confirms that the two systems have the same spatial resolution. The vorticity error δω is computed as the difference between MS and HDR vorticity. The results of table 4 show that the random error dominates over the mean bias error.

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The uncertainty at P is evaluated with the correlation statistics method (Wieneke 2015). Uncertainty propagation is done according to equation (42) using d  =  4 px, U  =  (U_u  +  U_v)/2 and ρ(2d)  =  0.45. The root-mean-square of the uncertainty is equal to U_u,rms  =  0.063 px and U_v,rms  =  0.064 px, which agrees very well with the error root-mean-square of 0.060 and 0.063, respectively. The calculation is repeated in the entire measurement domain in common between HDR and MS. The contours of figure 14 illustrate the comparison between the rms of the error and the uncertainty of the vorticity. Both uncertainty and error exhibit small variations within the considered domain, with values between 0.010 and 0.016 px/px. Again, the agreement between estimated uncertainty and error is very good.

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The time-resolved jet data are not suited for statistical analysis because the low effective number of independent samples (N_eff  =243, despite the total number of samples is N  =  8000) does not guarantee the statistical convergence of the results. Hence, to assess the uncertainty of statistical flow properties, the cavity flow data are used, where 2000 statistically independent velocity field are available.

Velocity data are extracted at a point P located close to the reattachment point; the turbulence intensity in P is equal to 22.0% of the free-stream velocity. The entire set of 2000 samples is divided into 100 independent subsets composed by 20 samples each. The statistical flow properties, namely time averages and Reynolds stresses, are computed from the subsets and compared with the value obtained with the entire set. Figure 15(left) shows the comparison between the time-averaged vertical velocity computed with the subsets of 20 samples and that evaluated from the entire set of 2000 samples. The uncertainty bars are evaluated with equation (15) and correspond to a theoretical confidence level of 68%. In most of the cases the results agree within the uncertainty of the measured mean velocity. To assess the accuracy of the uncertainty propagation formulae, the uncertainty coverage for different statistical quantities is computed and displayed in figure 15(right). The uncertainty coverage is defined as the number of samples for which the error is smaller than or equal to the estimated uncertainty. In case of Gaussian error distribution, the theoretical uncertainty coverage is about 68%. The results of figure 15(right) show the accuracy of the uncertainty propagation methodology: the uncertainty of the time-averaged quantities ($\bar{u}$ and $\bar{v}$) is accurate within 5%, whereas that of the Reynolds stresses is accurate within 10%.

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The effect of the number of samples on the accuracy of the statistical results is shown in figure 16. It is evident that the random uncertainty of the mean (figure 16(a)) is initially large and decreases with increasing sample size. In the entire range of sample sizes considered, the reference mean velocity is within the uncertainty bounds estimated with equation (15). Similarly, the normal Reynolds stress R_vv converges to the reference value with rate $1/\sqrt{N}$ (figure 16(b)). For low sample size (N  <  250), the measured R_vv overestimates the reference value due to the effect of spurious fluctuations by about 10%. A corrected value of R_vv is computed by subtracting the mean-square fluctuation: R_vv,corr  =  R_vv–$U_{\text{rms}}^{2}$. The uncertainty of uncorrected and corrected R_vv is computed via equations (18) and (23), respectively. The two uncertainties are the same within 1%, meaning that the uncertainty of R_vv is mainly due to statistical convergence rather than to the measurement uncertainty of u and v. For a correction of less than 1%, one would need at least 20 000 independent samples according to equation (24) before the uncertainty of the Reynolds stress decreases to the same level as the correction term $U_{\text{rms}}^{2}$. But a correction is nevertheless useful for low levels of Reynolds stress comparable to the uncertainties.

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The Reynolds shear stress R_uv is illustrated in figure 16(c). To compute the uncertainty ${{U}_{{{R}_{uv}}}}$, the cross-correlation coefficient between u and v is calculated: ρ_uv  =  0.41. The measured R_uv converges to the reference value with rate $1/\sqrt{N}$. As the estimated uncertainty, also the measurement error (difference between measured and reference value) decreases with increasing the sample size.