A posteriori uncertainty quantification of PIVbased pressure data
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Abstract
A methodology for a posteriori uncertainty quantification of pressure data retrieved from particle image velocimetry (PIV) is proposed. It relies upon the Bayesian framework, where the posterior distribution (probability distribution of the true velocity, given the PIV measurements) is obtained from the prior distribution (prior knowledge of properties of the velocity field, e.g., divergencefree) and the statistical model of PIV measurement uncertainty. Once the posterior covariance matrix of the velocity is known, it is propagated through the discretized Poisson equation for pressure. Numerical assessment of the proposed method on a steady Lamb–Oseen vortex shows excellent agreement with Monte Carlo simulations, while linear uncertainty propagation underestimates the uncertainty in the pressure by up to 30 %. The method is finally applied to an experimental test case of a turbulent boundary layer in air, obtained using timeresolved tomographic PIV. Simultaneously with the PIV measurements, microphone measurements were carried out at the wall. The pressure reconstructed from the tomographic PIV data is compared to the microphone measurements. Realizing that the uncertainty of the latter is significantly smaller than the PIVbased pressure, this allows us to obtain an estimate for the true error of the former. The comparison between true error and estimated uncertainty demonstrates the accuracy of the uncertainty estimates on the pressure. In addition, enforcing the divergencefree constraint is found to result in a significantly more accurate reconstructed pressure field. The estimated uncertainty confirms this result.
Keywords
Particle Image Velocimetry Particle Image Velocimetry Measurement Particle Image Velocimetry Data Gaussian Process Regression Tomographic Particle Image Velocimetry1 Introduction
Particle image velocimetry (PIV) is nowadays recognized as a reliable tool for the determination of the instantaneous static pressure in two or threedimensional flow fields (van Oudheusden 2013). The main advantage of the PIVbased pressure reconstruction with respect to conventional pressure probes (e.g., pressure transducers or microphones) is the instantaneous 2D or 3D pressure field determination, without the need of an expensive design and manufacturing of models where densely distributed sensors are installed (Ragni et al. 2011). PIVbased pressure measurements are made possible for applications where probes cannot be installed, such as microaerial vehicles (Tronchin et al. 2015). Additionally, they convey a quantitative visualization of the flow structures responsible for the aerodynamic forces acting on the model. Industrial tests typically require pressure information, for the evaluation of forces. Pressure fluctuations are important when dealing with fatigue studies and aeroacoustic source measurements (Haigermoser 2009; Liu and Katz 2013; Pröbsting et al. 2013).

the accuracy of the underlying velocity data;

the spatial and temporal resolutions, which affect the accuracy of velocity derivatives;

the approach used to evaluate the velocity material acceleration (Eulerian or Lagrangian approach);

the boundary conditions;

the pressure gradient integration procedure;

for PIVbased pressure reconstruction of planar PIV data, application of a 2D model to a 3D flow.
Several works have focused on evaluating the contribution of the above mentioned factors to the error of the reconstructed pressure. For example, de Kat and van Oudheusden (2012) derived an analytical expression of the error of the material acceleration for both Eulerian and Lagrangian approaches. In both approaches, the laser pulse separation \(\Updelta t\) has opposite effects on truncation and random error. Increasing \(\Updelta t\) increases truncation error, but decreases the relative random error. Furthermore, the error of the material acceleration is typically smaller when it is calculated with the Lagrangian approach. Violato et al. (2011) introduced a criterion for the selection of the minimum and maximum \(\Updelta t\) that accounts for the maximum allowed error on the material acceleration and the effect of the outofplane displacement of the particles. The authors found that the Lagrangian approach features a random error about 1.5 times smaller than that estimated for the Eulerian approach. The difference between the two approaches is less pronounced when looking at the resulting pressure field, due to the error suppression during the pressure gradient integration. Several schemes for the integration of the pressure gradient have been proposed, including the spacemarching integration (Baur and Köngeter 1999), the omnidirectional integration (Liu and Katz 2006) and the Poisson equation for pressure (Gurka et al. 1999). In de Kat and van Oudheusden (2010), a comparative assessment of pressure integration methods is conducted. Although the upshot was that the integration method has minor impact, and only the spacemarching approach is clearly inferior, their conclusions were based on statistical analysis and not on the instantaneous pressure uncertainty. In fact, no quantitative information about the uncertainty of computed pressure fields was provided in these early studies. In later studies by de Kat and van Oudheusden (2012) and de Kat and Ganapathisubramani (2013), quantitative information was provided. They used Gaussian vortices for their numerical verification. Charonko et al. (2010) reported that the error of the velocity field has a major influence on the accuracy of the reconstructed pressure. In their work, the pressure result became ‘unusable’ as soon as very small levels of random error were introduced. Extensive data postprocessing was required to reduce the velocity error and make accurate pressure reconstructions feasible. However, Murai et al. (2007) reported that a fairly simple smoothing of the velocity data is sufficient for accurate pressure results. Charonko et al. (2010) also investigated the effect of offaxis measurement and outofplane velocity. They found that schemes performing well in 2D conditions continued to do so for misalignments up to 30°. de Kat and van Oudheusden (2012) showed that the response to misalignments follows a cosine relation.
In turbulent flows, PIVbased pressure reconstruction presents two major challenges: the small magnitude of the pressure fluctuations and the large dynamic range (Ghaemi and Scarano 2013; Joshi et al. 2014). Ghaemi et al. (2012) conducted timeresolved PIV (TRPIV) measurements to investigate the pressure fluctuations in a turbulent boundary layer at \(Re_{\theta }=2400\). The agreement between the PIVbased pressure and that measured by a surface microphone was evaluated via the crosscorrelation coefficient, which was equal to 0.6. The probability density functions of the two signals agreed up to 3 kHz for tomographic PIV data and up to 1 kHz for planar data. Similarly Pröbsting et al. (2013) carried out the PIVbased pressure reconstruction in a turbulent boundary layer at \(Re_{\theta }=730\). The data were validated by comparison with a surface microphone and with DNS data. Also in this case, the PIVbased pressure signal showed fair agreement with the microphone signal, but the crosscorrelation coefficient between the two did not exceed 0.6. Despite the relatively high correlation coefficient achieved, those studies opened several doubts on the accuracy of the PIVbased pressure reconstruction. Acoustic emissions of the flow over a rectangular cavity were investigated by Koschatzky et al. (2011). The fluctuating pressure fields computed from TRPIV were used as input for the prediction of the sound emission. The authors found that the PIVbased pressure data and the direct pressure measurement at the cavity walls had a similar frequency spectrum. The main frequency tone and the first harmonic were correctly captured, but the amplitude was significantly underestimated. The PIV measurement uncertainty strongly influenced the reconstructed pressure at locations with weak pressure fluctuations.
From the discussion above, it emerges that PIVbased pressure is often highly inaccurate due to the many error sources that contribute to the total error. As a result, quantifying the uncertainty of the reconstructed pressure is of primary importance to the future use of this method. Furthermore, a posteriori uncertainty quantification could be used to inform the choice of experimental and postprocessing parameters, along with the pressure reconstruction approach, to maximize the precision of the pressure field. The present work discusses the mathematical framework for the uncertainty propagation from the velocity field to the pressure field. It is formulated from a Bayesian perspective, to allow the measurement uncertainty of the velocity field from PIV to be naturally combined with prior knowledge of the velocity field (e.g., divergencefree) (Wikle and Berliner 2007). Therefore, it is suitable for the quantification of the random uncertainty of pressure, which stems from the random uncertainty of the velocity. A number of a posteriori approaches have recently been proposed to quantify the PIV measurement uncertainty. Sciacchitano et al. (2015) have provided a comparison of these methods. The present framework can also propagate the uncertainty of the velocity spatial and temporal derivatives, when information on the latter is available. The uncertainty due to the application of a 2D model to a 3D flow—i.e., pressure from planar PIV—is not addressed here. We refer to Charonko et al. (2010) and de Kat and van Oudheusden (2012), who investigated the error resulting from applying planar PIV in a 3D flow.
The outline of this paper is as follows: Sect. 2 introduces the proposed methodology. Section 3 presents a numerical assessment with an analytical test case, where the proposed method is compared with Monte Carlo simulations and linear uncertainty propagation (JCGM 2008). Section 4 applies the method to the turbulent boundary layer experiment done by Pröbsting et al. (2013), where microphone measurements are available to assess the PIVbased pressure field and the usefulness of the confidence intervals obtained from the present uncertainty quantification method. Conclusions are drawn in Sect. 5.
2 Methodology
2.1 The velocity field uncertainty
2.1.1 The prior on the velocity field
Ultimately, one can either setup a prior covariance matrix for each velocity component independently, as was done by de Baar et al. (2014), or a single covariance matrix that constrains the velocity components according to mass conservation (the divergencefree or solenoidal constraint) (Azijli and Dwight 2015). The details can be found in these works. Whichever method is used, two parameters result that need to be determined, namely the prior variance \(\tau ^2\) from Eq. 4 and the correlation length \(\theta _k\) from Eq. 5. They can either be specified a priori or through a maximum likelihood optimization. Azijli and Dwight (2015) followed the a priori approach. This was possible because the PIV data had sufficient spatial resolution. de Baar et al. (2014) followed the optimization route since their PIV data had insufficient spatial resolution.
2.1.2 The PIV measurement uncertainty model
This situation where the pressure field is evaluated at the same locations where the velocity measurements are available is the most common (van Oudheusden 2013), but our framework allows generalization.
2.1.3 The posterior on the velocity field
2.2 Propagating uncertainties to the source term
The posterior distribution of the velocity (see Eqs. 7, 8) must be propagated through \({\mathbf{f}}\) to obtain the mean \({\mathbb{E}}\left( {\mathbf{f}}\right)\) and covariance \({\varSigma }\left( {\mathbf{f}}\right)\) of the source term. However, this is not straightforward since \({\mathbf{f}}\), derived from the Navier–Stokes equations, has a nonlinear dependence on the velocity. We consider two options to obtain the mean and covariance of the source term.
Equation 13 is valid irrespective of the distributions of a and b, since it is simply a rewrite of the definition of covariance. Equation 14 on the other hand is valid only if the distributions are Gaussian. Indeed, this is what we have done in the present work. The source term, however, is no longer Gaussian distributed.
It is important to state that this exact method is only possible if the material acceleration is evaluated with respect to a stationary reference frame, i.e., the Eulerian approach. If the Lagrangian approach is used, products arise with more than two random variables. Brown and Alexander (1991) derived general expressions for the covariance of a product of n random variables with a product of m random variables. However, these contain higherorder moments. If they cannot be determined, they need to be neglected, resulting in an approximation. The Monte Carlo method can straightforwardly be applied no matter which approach is used.
2.3 Uncertainty in the pressure field
The discretization error arising from solving the discrete system in Eq. 2 contributes to the overall uncertainty in the calculated pressure. This error can be reduced by interpolating the measurements onto a finer numerical mesh, which is possible with GPR, and then solving the (larger) resulting system. However, there is a moment where the interpolation error starts to dominate, thereby limiting the total error reduction. Fortunately, with GPR the interpolation is accompanied by confidence intervals through the posterior covariance matrix \({\varSigma }\left( {\mathbf{u}}{\mathbf{u}}_{\text {PIV}}\right)\). The interpolation error is therefore included in the pressure field uncertainty, provided that the system is solved on a sufficiently fine mesh.
2.4 A onedimensional example
We take six uniformly distributed measurement locations (\(m=6\)) but discretize the velocity at 41 points to illustrate the ability of GPR to interpolate the measurement points (\(n=41\)).
Measurement noise is included, with a variance of \(\sigma ^2= 2.5\times 10^{5}\). It is assumed that the measurement uncertainty at the locations are uncorrelated with each other, therefore \(R=\sigma ^2I\), where I is the identity matrix. Since there are 6 uniformly distributed measurement locations and the state contains 41 uniformly distributed points, the observation operator \(H\in {\mathbb{R}}^{6\times 41}\) is simply a Boolean matrix with \(H^{ij}=1\) at \(j=8i7\;(i=1,2,\ldots ,6)\) and \(H^{ij}=0\) everywhere else.
With all the relevant components defined, we can calculate the posterior mean and covariance with Eqs. 7 and 8, respectively. Figure 2b shows the results. Starting with the mean (top plot): combining the prior and measurements we obtain a posterior that agrees well with the true function evaluations. Proceeding to the bottom plot, the posterior variance is strongly decreased with respect to the prior variance from the bottom plot in Fig. 2a. Also, we can clearly see a decrease in the variance at the measurement locations and an increase in between. Indeed, the measurements are informative and the larger the distance from them, the larger the uncertainty in the reconstruction. Note that the variance is not decreasing to zero at the measurement locations, since we included measurement noise with a variance of \(\sigma ^2= 2.5\times 10^{5}\).
In the above discussion, we were able to calculate the error since we know the true pressure. In practice though, the true pressure is unknown and we aim to estimate it, which is of course the goal of the present work. The black line in Fig. 4 represents the estimated error resulting from solving Eq. 16 and shows that it is a good approximation of the true error, thereby illustrating the ability of the method to also include the interpolation error in the error estimate.
3 Numerical verification
4 Application to experimental data
The boundary conditions we impose are the same as applied by Pröbsting et al. (2013), namely Dirichlet at the top boundary and Neumann at all other boundaries. We use both the Eulerian and the Lagrangian approaches to evaluate the material acceleration. According to Violato et al. (2011), the Lagrangian approach should result in a more accurate reconstructed pressure field, since the boundary layer flow is a convection dominated type of flow. Whether this is indeed true should not only follow from comparison with the microphone measurement, but is also expected to be represented by a smaller uncertainty in the reconstructed pressure. To propagate the uncertainties, we carry out Monte Carlo simulations using 100 realizations. This is the number suggested by Houtekamer and Mitchell (1998) in the context of the Ensemble Kalman Filter, which is practically a Monte Carlo version of the Kalman Filter for nonlinear systems.
Following Pröbsting et al. (2013), Fig. 9 shows the pressure time series for a subset of the data. The red lines in the plots are the microphone signal. The results are normalized with the free stream dynamic pressure \(q_{\infty }\approx 60\) Pa. The fluctuations of the microphone signal are of the order of 2 % of the free stream dynamic pressure, which is around 1.2 Pa. The black lines in Fig. 9 are the PIVbased pressure signals. The results represent four cases. From top to bottom: without applying the prior knowledge of divergencefree velocity fields through the solenoidal filter and using the Eulerian approach; without the prior knowledge using the Lagrangian approach; including the solenoidal prior and using the Eulerian approach; including the solenoidal prior and using the Lagrangian approach. From the figure, we can already deduce the observations made by Violato et al. (2011), namely that the Lagrangian approach results in more accurate pressure reconstruction than the Eulerian approach, and the observations made by Azijli and Dwight (2015) that using the solenoidal prior improves the pressure reconstruction.
In addition to the PIVbased pressure signals, we have also plotted the estimated uncertainty. The bounds of the gray region are \(p_{\text {PIV}}\left( {\mathbf{x}}_{\text {mic}},t_i\right) \pm k\hat{\delta }_{\text {mic}}\left( t_i\right)\), where we have taken for the coverage factor \(k=1\), so an 68.3 % confidence level for a Gaussian distribution. We have not plotted the 95 % confidence level (\(k=1.96\)) since it would obscure the figures. Already from the plots, we can observe that the bounds indeed become tighter when switching from the Eulerian to the Lagrangian approach and when including the solenoidal prior.
Standard deviation of the true error, \(\text {std}(\delta )\) and the standard deviation of the approximated error averaged over the 1500 fields in time, \(\hat{\delta }_{\text {mic,avg}}\)
\(\text {std}\,\left( \delta \right) /q_{\infty }\)  \(\hat{\delta }_{\text {mic,avg}}/q_{\infty }\)  

No solenoidal prior (Eul)  0.022  0.021 
No solenoidal prior (Lag)  0.015  0.013 
Solenoidal prior (Eul)  0.012  0.015 
Solenoidal prior (Lag)  0.011  0.010 
Uncertainty effectiveness
Uncertainty effectiveness (%)  Theoretical value of uncertainty effectiveness (%)  

No solenoidal prior (Eul)  67; 93  68.3; 95 
No solenoidal prior (Lag)  61; 90  68.3; 95 
Solenoidal prior (Eul)  81; 98  68.3; 95 
Solenoidal prior (Lag)  68; 94  68.3; 95 
Skewness (skew) and kurtosis (kurt) of the velocity (vel) and pressure (pres) distribution, extracted from the microphone position in the PIV field
Skew (vel)  Skew (pres)  Kurt (vel)  Kurt (pres)  

No solenoidal prior (Eul)  0.034  −0.20  3.02  2.93 
No solenoidal prior (Lag)  0.034  −0.024  3.02  2.44 
Solenoidal prior (Eul)  0.034  −0.15  3.02  2.59 
Solenoidal prior (Lag)  0.034  0.15  3.02  2.89 
5 Conclusions
The present work proposed a methodology for the a posteriori quantification of the uncertainty of pressure data retrieved from PIV measurements. The approach relies upon the Bayesian framework, where the posterior distribution (probability distribution of the true velocity, given the PIV measurements) is obtained from the prior distribution (prior knowledge of the velocity, e.g., within a certain bound or divergencefree) and the distribution representing the PIV measurement uncertainty. Once the covariance matrix of the velocity is known, it is propagated through the Poisson equation. Provided the velocity uncertainty is Gaussian and the Eulerian approach is used, the uncertainty can be calculated exactly. Otherwise, Monte Carlo simulations can be used. Numerical assessment of the method on a steady Lamb–Oseen vortex showed excellent agreement of the propagated uncertainty with Monte Carlo simulations, while linear uncertainty propagation underestimates the uncertainty of the pressure by up to 30 %. The method was finally applied to an experimental test case of a turbulent boundary layer in air, obtained using timeresolved tomographic PIV. The pressure reconstructed from tomographic PIV data was compared to the surface measurement conducted by a microphone to determine the actual error of the former. The PIV uncertainty was quantified by applying a solenoidal filter to remove spurious divergence. In addition, spatial correlation of the PIV uncertainty was included. It was found that close to the wall, the velocity uncertainty increases, most likely due to the increased velocity gradient closer to the wall. This was observed to be propagated into the pressure field as well. The comparison between actual and approximated error showed the effectiveness of the proposed method for uncertainty quantification of pressure data from tomographic PIV experiments. The Lagrangian approach resulted in more accurate reconstructed pressure fields than the Eulerian approach, reducing the errors from approximately 2 % of the free stream dynamic pressure to approximately 1.5 %. Also, enforcing the divergencefree constraint was found to result in a more accurate reconstructed pressure field, eventually reducing the errors to 1 %. Both observations also followed from the uncertainty quantification through a decrease in the estimated uncertainty. In addition, the estimated errors were in excellent agreement with the actual error of the pressure.
Notes
Acknowledgments
We thank Dr. Stefan Pröbsting for providing the data set of the turbulent flat plate boundary layer in air.
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