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Using uncertainty to improve pressure field reconstruction from PIV/PTV flow measurements

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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

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Funding

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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Correspondence to Pavlos P. Vlachos.

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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

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