Session L24: Experimental Techniques: Data Analysis, Bias and Uncertainty

Three-Dimensional Velocity Field De-Noising using Modal Projection

PCMRI and Doppler ultrasound are common modalities for imaging velocity fields inside the body (e.g. blood, air, etc) and PCMRI is increasingly being used for other fluid mechanics applications where optical imaging is difficult. This type of imaging is typically applied to internal flows, which are strongly influenced by domain geometry. While these technologies are evolving, it remains that measured data is noisy and boundary layers are poorly resolved. We have developed a boundary modal analysis method to de-noise 3D velocity fields such that the resulting field is divergence-free and satisfies no-slip/no-penetration boundary conditions. First, two sets of divergence-free modes are computed based on domain geometry. The first set accounts for flow through ``truncation boundaries'', and the second set of modes has no-slip/no-penetration conditions imposed on all boundaries. The modes are calculated by minimizing the velocity gradient throughout the domain while enforcing a divergence-free condition. The measured velocity field is then projected onto these modes using a least squares algorithm. This method is demonstrated on CFD simulations with artificial noise. Different degrees of noise and different numbers of modes are tested to reveal the capabilities of the approach.   

A vorticity transport model to restore spatial gaps in velocity data

Often measurements of velocity data do not have full spatial coverage in the probed domain or near boundaries. These gaps can be due to missing measurements or masked regions of corrupted data. These gaps confound interpretation, and are problematic when the data is used to compute Lagrangian or trajectory-based analyses. Various techniques have been proposed to overcome coverage limitations in velocity data such as unweighted least square fitting, empirical orthogonal function analysis, variational interpolation as well as boundary modal analysis. In this talk, we present a vorticity transport PDE to reconstruct regions of missing velocity vectors. The transport model involves both nonlinear anisotropic diffusion and advection. This approach is shown to preserve the main features of the flow even in cases of large gaps, and the reconstructed regions are continuous up to second order. We illustrate results for high-frequency radar (HFR) measurements of the ocean surface currents as this is a common application of limited coverage. We demonstrate that the error of the method is on the same order of the error of the original velocity data. In addition, we have developed a web-based gateway for data restoration, and we will demonstrate a practical application using available data.   

Uncertainty of Reynolds Stresses from PIV Measurements

Particle Image Velocimetry measurements of fluid velocity have poor dynamic range and therefore have significant random error. Over the last seven years, excellent progress has been made toward estimating the random uncertainty of individual PIV vectors automatically in an a posteriori manner. In fact, two leading commercial PIV codes now have uncertainty estimation as an option in the vector calculation. The impact of random errors on the time-average of velocity is only to increase the number of data points required for convergence. However, other important fluid dynamics quantities, such as Reynolds normal stresses, will contain a bias error due to random errors in the vectors used to compute the quantity. If one has an accurate estimate of the instantaneous uncertainties, this bias may be estimated and removed. However, recent studies have shown that the performance of uncertainty estimation schemes varies depending on many factors and that no scheme is accurate in every case. In this presentation, we will discuss the impact of inaccurate estimation of instantaneous random uncertainty on the uncertainty of Reynolds stress.   

Uncertainty Quantification from Measures of Divergence in 2D PIV Data

Particle image velocimetry (PIV) is subject to various sources of uncertainty that can lead to erroneous velocities in the derived vector fields. In recent years, significant effort has been focused on developing methods to quantify this uncertainty. In general, these methods calculate uncertainty from the raw PIV images and the quality of the cross correlation. This project has been initiated to develop a method for quantitative data quality assessment using vector fields alone. This approach utilizes the fact that for incompressible flows, erroneous vectors will result in non-zero divergence. The presented work focuses on quantifying the relationship between uncertainty and divergence. Planar PIV measurements were taken in the wake of a NACA 0015 airfoil at four angles of attack and uncertainty was calculated using the correlation statistics method implemented in the LaVision DaVis software. Correlations were then made between the divergence and uncertainty fields through methods of cross-correlation and covariance. A correlation of approximately 50 percent was found for the raw fields; however, this value is sensitive to data filtering and noise floor levels.   

The Uncertainty of Volume-Flow Rate Inflow/Outflow Measurement By Integrating PIV Velocity Fields

The purpose of this work is to assess the performance of non-intrusive volume-flow rate measurements acquired by various Particle Image Velocimetry (PIV) techniques. Both two-component (2C) and stereo (3C) PIV data sets were acquired at the exits of a high aspect ratio rectangular and round planar nozzles for turbulent flow rates of Reynolds numbers between 10,000 and 100,000. The PIV data sets were processed numerous ways by systematically changing the algorithms and parameters. The time-averaged results were then spatially integrated across the planar nozzle exit planes and compared to a calibrated flow meter and Laser Doppler Velocimetry (LDV) data. The PIV measurement performance metrics that are investigated in this work include uncertainty, calculation time, and volume-flow rate deviation. Recommendations for each method are developed and listed with potential drawbacks. The accuracy of the measurement was found to be a weak function of the Reynolds number of the flow. 2C-PIV was found to underestimate volume-flow rate by 2-4% depending on the integration scheme and stereo PIV underestimated volume-flow rate by 2%.   

Load estimation from planar PIV measurement in vortex dominated flows.

Control volume-based loading estimates are employed on experimental and synthetic numerical planar Particle Image Velocimetry (PIV) data of a stationary cylinder and a cylinder undergoing one degree-of-freedom (1DOF) Vortex Induced Vibration (VIV). The results reveal the necessity of including out of plane terms, identified from a general formulation of the control volume momentum balance, when evaluating loads from planar measurements in three-dimensional flows. Reynolds stresses from out of plane fluctuations are shown to be significant for both instantaneous and mean force estimates when the control volume encompasses vortex dominated regions. For planar measurement, invoking a divergence-free assumption allows accurate estimation of half the identified terms. Towards evaluating the fidelity of PIV-based loading estimates for obtaining the forcing function unobtrusively in VIV experiments, the accuracy of the control volume-based loading methodology is evaluated using the numerical data with synthetically generated experimental PIV error, and a comparison is made between experimental PIV-based estimates and simultaneous force balance measurements.   

3D flow effects on measuring turbulence statistics using 2D PIV

Homogeneous {\&} isotropic turbulence (HIT) with no mean flow is the simplest type of turbulent flow which can be used to study various phenomena. Although HIT is inherently three dimensional in nature, various turbulence statistics can be measured with 2D PIV utilizing various assumptions. In this study, the loss of tracer particle pairs due to out-of-plane motion, and the effect it has on statistics such as turbulence kinetic energy, dissipation rate, and velocity correlations is investigated. Synthetic PIV images created from HIT direct numerical simulation (DNS) data are utilized to quantify this effect. We estimate the out-of-plane error by adjusting parameters such as PIV time interval, interrogation window size, and particle size. This information can be utilized to optimize experimental parameters when examining 3D turbulence via 2D PIV.   

Determining Stationary Episodes in Field Observations

Time-averaged turbulence statistics from field observations are required to educe theoretical relationships and to validate numerical simulations. Meaningful time averages rely upon episodes with stationary mean values. A novel approach to determine both the occurrence and duration of stationary episodes within time series is constructed. The reverse arrangement test, a classical technique providing robust measure of mean trends, is chosen as the basic statistical operation. The probability distributions of the starting and ending points of stationary intervals are used to determine (i) the nonstationary location at which a time period should be split into two sub-periods, and (ii) nonstationary samples that should be discarded from further analysis of time-averaged statistics. The approach provides an efficient technique to analyze long-term datasets, and is capable of relating data sampled at multiple locations. Applying the approach to data obtained within and above a walnut orchard canopy during the Canopy Horizontal Array Turbulent Study yields a clean relationship between the canopy-top mean wind and mean shear stress. Using this approach to determine stationary episodes is also essential for accurately determining a sonic anemometer’s coordinate system in the field.   

Discovering Hidden Controlling Parameters using Data Analytics and Dimensional Analysis

Dimensional Analysis is a powerful tool, one which takes a priori information and produces important simplifications. However, if this a priori information -- the list of relevant parameters -- is missing a relevant quantity, then the conclusions from Dimensional Analysis will be incorrect. In this work, we present novel conclusions in Dimensional Analysis, which provide a means to detect this failure mode of missing or hidden parameters. These results are based on a restated form of the Buckingham Pi theorem that reveals a ridge function structure underlying all dimensionless physical laws. We leverage this structure by constructing a hypothesis test based on sufficient dimension reduction, allowing for an experimental data-driven detection of hidden parameters. Both theory and examples will be presented, using classical turbulent pipe flow as the working example. Keywords: experimental techniques, dimensional analysis, lurking variables, hidden parameters, buckingham pi, data analysis   

Error Propagation dynamics: from PIV-based pressure reconstruction to vorticity field calculation

Noninvasive data from velocimetry experiments (e.g., PIV) have been used to calculate vorticity and pressure fields. However, the noise, error, or uncertainties in the PIV measurements would eventually propagate to the calculated pressure or vorticity field through reconstruction schemes. Despite the vast applications of pressure and/or vorticity field calculated from PIV measurements, studies on the error propagation from the velocity field to the reconstructed fields (PIV-pressure $^{\mathrm{[1-3]}}$ and PIV-vorticity $^{\mathrm{[4-5]}})$ are few. In the current study, we break down the inherent connections between PIV-based pressure reconstruction and PIV-based vorticity calculation. The similar error propagation dynamics, which involve competition between physical properties of the flow and numerical errors from reconstruction schemes, are found in both PIV-pressure and PIV-vorticity reconstructions. [1] McClure and Yarusevych, 2017,~Exp. Fluids,~58(5). [2] Pan \textit{et al}., 2016, Meas. Sci. Technol.,~27(8). [3] Charonko \textit{et al.}, 2010, Meas. Sci. Technol. 21(10). [4] Luff \textit{et al.}, 1999,~Exp. Fluids,~26(1). [5] Lecuona \textit{et al.}, 1998,~J. Vis.,~1(2).   

Robust estimation of the integral scale for quantifying uncertainty of the sample mean from non-independent velocity data

Using large data sets, we evaluate statistical bootstrapping schemes for approximating uncertainty in the sample mean of highly correlated velocity field measurements. Interest in time-resolved velocity field data has led to sampling rates high enough that non-independent samples are commonplace. Uncertainty of the sample mean collected from stationary but correlated data is given by $s/ \sqrt{N_{eff}}$, where $s$ is the standard deviation of the samples, and $N_{eff}$ is the "effective" number of samples, that is, the number of samples $N$ divided by twice the integral time scale $T_u$. We can approximate $T_u$ using of a sum of auto-correlation coefficients, but it is necessary to truncate the sum at a prescribed lag $K$. This lag parameter $K$ is equivalent to a bootstrapping parameter in statistics, and we can optimize selection of $K$ using techniques from the bootstrapping methodology. With highly resolved data from laminar and turbulent velocity field measurements we will evaluate different strategies for this statistical bootstrap optimization.   

Uncertainty Propagation In The Singular Value Decomposition Of Measured Data

Singular value decomposition (SVD) is a well-known mathematical tool that can be used to decompose an ensemble of velocity field data into spatiotemporal modes that may reveal coherent flow structures. Proper orthogonal decomposition (POD) is a special case of the SVD commonly used when the data are uncorrelated in time (such as in a turbulent flow) or for building surrogate models. Although the SVD and POD have been widely used in fluid mechanics, Epps and Techet (2010, ExpFluids 48:355–367) were among the first to consider how experimental error affects the results of the SVD. This talk briefly reviews the work of that paper and provides mathematically-rigorous bounds on the errors in the computed singular values and spatiotemporal mode shapes. Using a constructed dataset with known, applied error as an example, the process to (i) determine the root mean square measurement error and (ii) determine “error bars” for the singular values and vectors is demonstrated. This process is then applied to measured data with unknown error.