Abstract
A novel pressure reconstruction method is proposed to use the uncertainty information to improve the instantaneous pressure fields from velocity fields measured using particle image velocimetry (PIV) or particle tracking velocimetry (PTV). First, the pressure gradient fields are calculated from velocity fields, while the local and instantaneous pressure gradient uncertainty is estimated from the velocity uncertainty using a linear-transformation-based algorithm. The pressure field is then reconstructed by solving an overdetermined linear system which involves the pressure gradients and boundary conditions. This linear system is solved with generalized least squares (GLS) which incorporates the previously estimated variances and covariances of the pressure gradient errors as inverse weights to optimize the reconstructed pressure field. The method was validated with synthetic velocity fields of a 2D pulsatile flow, and the results show significantly improved pressure accuracy. The pressure error reduction by GLS was 50% with 9.6% velocity errors and 250% with 32.1% velocity errors compared to the existing baseline method of solving the pressure Poisson equation (PPE). The GLS was more robust to the velocity errors and provides greater improvement with spatially correlated velocity errors. For experimental validation, the volumetric pressure fields were evaluated from the velocity fields measured using 3D PTV of a laminar pipe flow with a Reynolds number of 630 and a transitional pipe flow with a Reynolds number of 3447. The GLS reduced the median absolute pressure errors by as much as 96% for the laminar pipe flow compared to PPE. The mean pressure drop along the pipe predicted by GLS was in good agreement with the empirical estimation using Darcy–Weisbach equation for the transitional pipe flow.
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Abbreviations
- x, y, z :
-
Spatial coordinates
- \( \Delta x \), \( \Delta y \) :
-
Grid sizes
- \( \Delta t \) :
-
Time interval between consecutive frames
- u, v, w :
-
Velocity components
- \( \varvec{u} \) :
-
Velocity vector
- \( \varvec{u}_{\varvec{m}} \) :
-
Measured velocity
- \( \varvec{u}_{\varvec{t}} \) :
-
True velocity
- \( \varvec{\epsilon}_{\varvec{u}} \) :
-
Error in measured velocity
- \( \Sigma _{u} \) :
-
Covariance matrix of velocity errors
- \( \varvec{\sigma}_{\varvec{u}} \) :
-
Standard deviation of velocity errors
- \( \rho_{u1,u2} \) :
-
Auto-correlation coefficient between velocity errors
- \( {\text{Cov}}_{u1,u2} \) :
-
Covariance between velocity errors
- p :
-
Pressure
- \( \epsilon_{p} \) :
-
Error in the reconstructed pressure
- \( p_{\text{ref}} \) :
-
Reference pressure
- \( \epsilon_{{p_{\text{ref}} }} \) :
-
Error in the reference pressure
- \( \varvec{p}_{{\varvec{grad},\varvec{u}}} \) :
-
Pressure gradient calculated from the measured velocity
- \( \varvec{p}_{{\varvec{grad},\varvec{t}}} \) :
-
True pressure gradient
- \( \varvec{\epsilon}_{{\nabla \varvec{p}}} \) :
-
Error in the calculated pressure gradient
- \( \varvec{\sigma}_{{\nabla \varvec{p}}} \) :
-
Standard deviation of pressure gradient errors
- \( \Sigma _{\nabla p} \) :
-
Covariance matrix of pressure gradient errors
- \( \epsilon_{pgradx} \) :
-
Streamwise pressure gradient errors
- \( \sigma_{pgradx} \) :
-
Standard deviation of streamwise pressure gradient errors
- \( \rho_{pgradx} \) :
-
Autocorrelation coefficients between streamwise pressure gradient errors
- \( \rho \) :
-
Fluid density
- \( \mu \) :
-
Fluid dynamic viscosity
- STD:
-
Standard deviation
- RMS:
-
Root-mean-square
References
Azijli I, Dwight RP (2015) Solenoidal filtering of volumetric velocity measurements using Gaussian process regression. Exp Fluids. https://doi.org/10.1007/s00348-015-2067-7
Azijli I, Sciacchitano A, Ragni D et al (2016) A posteriori uncertainty quantification of PIV-based pressure data. Exp Fluids. https://doi.org/10.1007/s00348-016-2159-z
Bhattacharya S, Charonko JJ, Vlachos PP (2017) Stereo-particle image velocimetry uncertainty quantification. Measur Sci Technol 28:015301. https://doi.org/10.1088/1361-6501/28/1/015301
Bhattacharya S, Charonko JJ, Vlachos PP (2018) Particle image velocimetry (PIV) uncertainty quantification using moment of correlation (MC) plane. Measur Sci Technol 29:115301. https://doi.org/10.1088/1361-6501/aadfb4
Bhattacharya S, Vlachos PP (2019) Volumetric particle tracking velocimetry (PTV) uncertainty quantification. arXiv:1911.12495
Björck A (1996) Numerical methods least squares squares. Society for Industrial and Applied Mathematics, Philadelphia
Charonko JJ, King CV, Smith BL, Vlachos PP (2010) Assessment of pressure field calculations from particle image velocimetry measurements. Measur Sci Technol 21:105401. https://doi.org/10.1088/0957-0233/21/10/105401
Charonko JJ, Vlachos PP (2013) Estimation of uncertainty bounds for individual particle image velocimetry measurements from cross-correlation peak ratio. Measur Sci Technol 24:065301. https://doi.org/10.1088/0957-0233/24/6/065301
Dabiri JO, Bose S, Gemmell BJ et al (2014) An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements. J Exp Biol 217:331–336. https://doi.org/10.1242/jeb.092767
de Kat R, Van Oudheusden BW (2012) Instantaneous planar pressure determination from PIV in turbulent flow. Exp Fluids 52:1089–1106. https://doi.org/10.1007/s00348-011-1237-5
de Kat R, van Oudheusden BW, Scarano F (2009) Instantaneous pressure field determination around a square-section cylinder using time-resolved stereo-PIV. In: 39th AIAA fluid dynamics conference, pp 1–10
Fujisawa N, Tanahashi S, Srinivas K (2005) Evaluation of pressure field and fluid forces on a circular cylinder with and without rotational oscillation using velocity data from PIV measurement. Measur Sci Technol 16:989–996. https://doi.org/10.1088/0957-0233/16/4/011
Gesemann S, Huhn F, Schanz D, Schröder A (2016) From noisy particle tracks to velocity, acceleration and pressure fields using B-splines and penalties. In: 18th international symposium on the application of laser and imaging techniques to fluid mechanics
Ghaemi S, Ragni D, Scarano F (2012) PIV-based pressure fluctuations in the turbulent boundary layer. Exp Fluids 53:1823–1840. https://doi.org/10.1007/s00348-012-1391-4
Huhn F, Schanz D, Gesemann S, Schröder A (2016) FFT integration of instantaneous 3D pressure gradient fields measured by Lagrangian particle tracking in turbulent flows. Exp Fluids 57:151. https://doi.org/10.1007/s00348-016-2236-3
Jeon YJ, Gomit G, Earl T et al (2018) Sequential least-square reconstruction of instantaneous pressure field around a body from TR-PIV. Exp Fluids 59:27. https://doi.org/10.1007/s00348-018-2489-0
Li XS (2005) An overview of SuperLU: algorithms, implementation, and user interface. ACM Trans Math Softw 31:302–325
Liu X, Katz J (2006) Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Exp Fluids 41:227–240. https://doi.org/10.1007/s00348-006-0152-7
Maas HG, Gruen A, Papantoniou D (1993) Particle tracking velocimetry in three-dimensional flows—Part 1. Photogrammetric determination of particle coordinates. Exp Fluids 15:133–146. https://doi.org/10.1007/BF00190953
McKeon B, Engler R (2007) Pressure measurement systems. In: Springer Handbook of Experimental Fluid Mechanics. Springer-Verlag, Berlin, Germany, pp 179–214
Moody LF (1944) Friction factor for pipe flow. Trans ASME 66:671–684
Neeteson NJ, Rival DE (2015) Pressure-field extraction on unstructured flow data using a Voronoi tessellation-based networking algorithm: a proof-of-principle study. Exp Fluids 56:44. https://doi.org/10.1007/s00348-015-1911-0
Pan Z, Whitehead J, Thomson S, Truscott T (2016) Error propagation dynamics of PIV-based pressure field calculations: how well does the pressure Poisson solver perform inherently? Measur Sci Technol 27:084012. https://doi.org/10.1088/0957-0233/27/8/084012
Rao RC (1973) Linear Statistical Inference and Its Applications, 2nd ed., Wiley, New York
Schanz D, Gesemann S, Schröder A (2016) Shake-The-Box: Lagrangian particle tracking at high particle image densities. Exp Fluids 57:70. https://doi.org/10.1007/s00348-016-2157-1
Schiavazzi DE, Nemes A, Schmitter S, Coletti F (2017) The effect of velocity filtering in pressure estimation. Exp Fluids 58:50. https://doi.org/10.1007/s00348-017-2314-1
Schneiders JFG, Dwight RP, Scarano F (2014) Time-supersampling of 3D-PIV measurements with vortex-in-cell simulation. Exp Fluids 55:1692. https://doi.org/10.1007/s00348-014-1692-x
Schneiders JFG, Scarano F (2016) Dense velocity reconstruction from tomographic PTV with material derivatives. Exp Fluids 57:139. https://doi.org/10.1007/s00348-016-2225-6
Schneiders JFG, Sciacchitano A (2017) Track benchmarking method for uncertainty quantification of particle tracking velocimetry interpolations. Measur Sci Technol 28:065302. https://doi.org/10.1088/1361-6501/aa6a03
Sciacchitano A, Neal DR, Smith BL et al (2015) Collaborative framework for PIV uncertainty quantification: comparative assessment of methods. Meas Sci Technol 26:074004. https://doi.org/10.1088/0957-0233/26/7/074004
Sciacchitano A, Wieneke B (2016) PIV uncertainty propagation. Measur Sci Technol 27:084006. https://doi.org/10.1088/0957-0233/27/8/084006
Soloff SM, Adrian RJ, Liu ZC (1997) Distortion compensation for generalized stereoscopic particle image velocimetry. Measur Sci Technol 8:1441–1454. https://doi.org/10.1088/0957-0233/8/12/008
Timmins BH, Wilson BW, Smith BL, Vlachos PP (2012) A method for automatic estimation of instantaneous local uncertainty in particle image velocimetry measurements. Exp Fluids 53:1133–1147. https://doi.org/10.1007/s00348-012-1341-1
Tronchin T, David L, Farcy A (2015) Loads and pressure evaluation of the flow around a flapping wing from instantaneous 3D velocity measurements. Exp Fluids 56:7. https://doi.org/10.1007/s00348-014-1870-x
van Gent P, Michaelis D, van Oudheusden BW et al (2017) Comparative assessment of pressure field reconstructions from particle image velocimetry measurements and Lagrangian particle tracking. Exp Fluids 58:33. https://doi.org/10.1007/s00348-017-2324-z
Van Gent PL, Schrijer FFJ, Van Oudheusden BW (2018a) Assessment of the pseudo-tracking approach for the calculation of material acceleration and pressure fields from time-resolved PIV: Part I. Error propagation. Measur Sci Technol 29:045204. https://doi.org/10.1088/1361-6501/aaa0a5
Van Gent PL, Schrijer FFJ, Van Oudheusden BW (2018b) Assessment of the pseudo-tracking approach for the calculation of material acceleration and pressure fields from time-resolved PIV: Part II. Spatio-temporal filtering. Measur Sci Technol 29:045206. https://doi.org/10.1088/1361-6501/aaab84
van Oudheusden BW (2013) PIV-based pressure measurement. Measur Sci Technol 24:032001. https://doi.org/10.1088/0957-0233/24/3/032001
Violato D, Moore P, Scarano F (2011) Lagrangian and Eulerian pressure field evaluation of rod-airfoil flow from time-resolved tomographic PIV. Exp Fluids 50:1057–1070. https://doi.org/10.1007/s00348-010-1011-0
Wang CY, Gao Q, Wei RJ et al (2017) Spectral decomposition-based fast pressure integration algorithm. Exp Fluids 58:84. https://doi.org/10.1007/s00348-017-2368-0
Wang Z, Gao Q, Wang C et al (2016) An irrotation correction on pressure gradient and orthogonal-path integration for PIV-based pressure reconstruction. Exp Fluids 57:104. https://doi.org/10.1007/s00348-016-2189-6
Wieneke B (2008) Volume self-calibration for 3D particle image velocimetry. Exp Fluids 45:549–556. https://doi.org/10.1007/s00348-008-0521-5
Wieneke B (2017) PIV anisotropic denoising using uncertainty quantification. Exp Fluids 58:94. https://doi.org/10.1007/s00348-017-2376-0
Wieneke B (2013) Iterative reconstruction of volumetric particle distribution. Measur Sci Technol 24:024008. https://doi.org/10.1088/0957-0233/24/2/024008
Xue Z, Charonko JJ, Vlachos PP (2014) Particle image velocimetry correlation signal-to-noise ratio metrics and measurement uncertainty quantification. Measur Sci Technol 25:115301. https://doi.org/10.1088/0957-0233/25/11/115301
Xue Z, Charonko JJ, Vlachos PP (2015) Particle image pattern mutual information and uncertainty estimation for particle image velocimetry. Measur Sci Technol 26:074001. https://doi.org/10.1088/0957-0233/26/7/074001
Zhang J, Brindise MC, Rothenberger S et al (2020) 4D flow MRI pressure estimation using velocity measurement-error based weighted least-squares. IEEE Trans Med Imaging 39:1668–1680. https://doi.org/10.1109/TMI.2019.2954697
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Funding was provided by National Institutes of Health (US) (Grant No. NIH R21 NS106696) and National Science Foundation (Grant No. NSF 1706474).
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Zhang, J., Bhattacharya, S. & Vlachos, P.P. Using uncertainty to improve pressure field reconstruction from PIV/PTV flow measurements. Exp Fluids 61, 131 (2020). https://doi.org/10.1007/s00348-020-02974-y
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DOI: https://doi.org/10.1007/s00348-020-02974-y