Error reduction for time-resolved PIV data based on Navier–Stokes equations

Abstract

The post-processing of the measured velocity in particle image velocimetry (PIV) is a critical step in reducing error and predicting missing information of the flow field. In this work, time-resolved PIV data are incorporated with the incompressible Navier–Stokes (N–S) equations to reduce the measurement error and improve the accuracy. A pressure correction scheme (PCS) based on the projection method is adopted to solve the N–S equations, and an optimization algorithm is introduced to balance the fidelity between the PIV data and the numerical solutions. The PCS for PIV data, called PIV–PCS, cannot only reduce the errors in the velocity divergence and the curl of the pressure gradient but also ensure that the flow field satisfies the dynamic constraints imposed by the momentum equation. An important weight coefficient s that balances the level of the velocity modification with the residual of the governing equation is defined and numerically assessed. A method for optimizing the value of s is provided. The new approach is evaluated by two time-resolved PIV experiments: one on the 2D wake flow of a circular cylinder at low Reynolds number and one on tomographic PIV for the 3D wake flow of a hemisphere at high Reynolds number. All the numerical assessments and experimental applications are compared with the divergence-free smoothing (DFS) method. The results indicate that the presented PIV–PCS method is superior to the DFS method in terms of reducing the measurement error and recovering the real physical flow structures.

Appendix

To inspect the suitability of $\alpha$ = 10 at lower sampling frequencies, the 2D NRMSE map and the time history of the RMS difference under a sampling frequency of 50 Hz are given in Figs. 15 and 16, respectively. The illustrations in Figs. 15 and 16 are the same as that in Figs. 4 and 5, except certain values are different. The error in Fig. 15 is higher than that in Fig. 4 because of the larger time separation. The error increases with the noise level and still presents a minimum in the region of $0.7 \le s \le 0.8$. The optimal s computed by $\alpha = 10$ is equal to 0.7, which is very close to the result from the figure. In Fig. 16, the time evolution of the error for u and v shows a different behavior. The PIV–PCS error of $s=0.7$ for horizontal u overlaps with that of $s = 0.97$; however, the error in the vertical component v of $s=0.7$ is substantially smaller than that of $s = 0.97$. The plot of $s = 1.0$ is not shown due to the large error. From these figures, the adoption of $\alpha$ = 10 is found to be reasonable for estimating the optimal s.

Fig. 15

The 2D contour map of the NRMSE as a function of noise level ($\epsilon _u$) and s. The parameter s is presented in the range of $0.5\le s\le 1.0$. a BCs–Raw, b BCs–POD. Note that the range of the colormap is different

Fig. 16

The time evolution of the RMS difference between numerical simulation and PIV–PCS for the streamwise velocity (a) and vertical velocity (b). The black, purple, green and red lines represent the results of artificial noise and PIV–PCS with s = 0.5, 0.7 and 0.97. The noise level $\epsilon _u$ is set to $5\%$. The range of the y-coordinate is different