Divergence-free smoothing for volumetric PIV data

Abstract

This paper proposes a divergence-free smoothing (DFS) method for the post-process of volumetric particle image velocimetry (PIV) data, which can smooth out noise and divergence error at the same time. The method is a combination of the penalized least squares regression and the divergence corrective scheme (DCS), employing the generalized cross-validation method to automatically determine the best smoothing parameter. By introducing a weight-changing algorithm similar to the all-in-one method, a robust version of DFS can simultaneously deal with vector validation, replacement of outliers and missing vectors, smoothing, and zero-divergence correction of the velocity field. Direct numerical simulation data of turbulent channel flow (Johns Hopkins Turbulence Databases) added with artificial noise, outliers and missing vectors are used to test the accuracy of DFS. The results show that DFS can smooth the velocity field to divergence-free and performs better than the all-in-one method, DCS and some other available conventional processing methods for post-process of velocity field, especially in dealing with clustered outliers and missing vectors. A block DFS is suggested to process large velocity field to save both time and memory. Tests on tomographic PIV data validate the effectiveness of DFS on improving both flow statistics and flow visualization.

Appendices

Appendix 1: Specific forms of ${\textbf{A }}$ and ${\textbf{M}}$

Assume that the l-th ($l=1, 2, 3$) component of velocity at the position (i, j, k) is arranged as the $(i+(j-1)n_x+(k-1)n_xn_y+(l-1)n)$-th element of ${\textbf{U }}_{\text {exp}}$ or ${\textbf{U }}_{\text {s}}$. Then corresponding divergence operator ${\textbf{A }}$ and ${\textbf{M}}$ are calculated by the Eqs. (39)–(42), noting that the symbol $\otimes$ denotes the Kronecker product between two matrices.

$$\begin{aligned} {\textbf{A }}&=\left[ \frac{1}{\Delta x}{\textbf{I }}_{n_z}\otimes {\textbf{I }}_{n_y}\otimes {\textbf {D}}_{n_x},\right.\\&\left.\quad \frac{1}{\Delta y} {\textbf{I }}_{n_z}\otimes {\textbf {D}}_{n_y}\otimes {\textbf{I }}_{n_x},\frac{1}{\Delta z} {\textbf {D}}_{n_z}\otimes {\textbf{I }}_{n_y}\otimes {\textbf{I }}_{n_x}\right] , \end{aligned}$$

(39)

where ${\textbf{I }}_{n_x}$, ${\textbf{I }}_{n_x}$ and ${\textbf{I }}_{n_z}$ are identical matrices and ${\textbf {D}}_{n_x}$, ${\textbf {D}}_{n_x}$ and ${\textbf {D}}_{n_z}$ take the following form

$$\begin{aligned} {\textbf {D}}_m= & {} \begin{bmatrix} -1&1\\ -1/2&0&1/2\\&-1/2&0&1/2\\&&\dots&\\&&-1&1 \end{bmatrix}_{m\times m}(m=n_x, n_y \ {\text {or}} \ n_z). \end{aligned}$$

(40)

$$\begin{aligned} {\textbf{M}}= & {} \frac{1}{\Delta x^2}{\textbf{I }}_{3}\otimes {\textbf{I }}_{n_z}\otimes {\textbf{I }}_{n_y}\otimes {\textbf{N}}_{n_x}^2+ \frac{1}{\Delta y^2}{\textbf{I }}_{3}\otimes {\textbf{I }}_{n_z}\otimes {\textbf{N}}_{n_y}^2\otimes {\textbf{I }}_{n_x}\nonumber \\&+\,\frac{1}{\Delta z^2}{\textbf{I }}_{3}\otimes {\textbf{N}}_{n_z}^2\otimes {\textbf{I }}_{n_y}\otimes {\textbf{I }}_{n_x} +\frac{2}{\Delta x \Delta y} {\textbf{I }}_{3}\otimes {\textbf{I }}_{n_z}\\&\otimes \,{\textbf{N}}_{n_y}\otimes {\textbf{N}}_{n_x}+\frac{2}{\Delta y \Delta z} {\textbf{I }}_{3}\otimes {\textbf{N}}_{n_z}\\&\otimes\, {\textbf{N}}_{n_y}\otimes {\textbf{I }}_{n_x} +\frac{2}{\Delta x \Delta z} {\textbf{I }}_{3}\otimes {\textbf{N}}_{n_z}\otimes {\textbf{I }}_{n_y}\otimes {\textbf{N}}_{n_x}, \end{aligned}$$

(41)

where ${\textbf{N}}_{n_x}$, ${\textbf{N}}_{n_x}$ and ${\textbf{N}}_{n_z}$ take the following form.

$$\begin{aligned} {\textbf{ N}}_m=\begin{bmatrix} -1&1\\ 1&-2&1\\&1&-2&1\\&&\dots&1\\&&1&-1 \end{bmatrix}_{m\times m}(m=n_x, n_y \ {\text {or}}\ n_z). \end{aligned}$$

(42)

The Eq. (41) is a generalization of the work of Buckley (1994), who detailed the form of ${\textbf{M}}$ for the smoothing of two-dimensional scalar data.

Appendix 2: Numerical test for GCV

To validate the performance of the GCV method on estimating smoothing parameter, a test based on 100 DNS velocity fields with different noise levels is performed to get the average results. Details of the testing DNS data and the added artificial noise are introduced in Sect. 3, where more tests of DFS are performed on the same DNS data. To find the best smoothing parameter s directly in this test, DFS using different value of s (from 0.02 to 1) is scanned to smooth the noisy velocity field, with which the estimation of s by GCV method is compared. The relative smoothing error is defined as the norm of the difference between the DFS-processed field and the original DNS field, normalized by the norm of the original DNS field with the free-stream velocity subtracted [Eq. (36) in Sect. 3.1]. Errors of DFS using different s to smooth velocity field with different noise levels are contoured in Fig. 18. The best s marked in Fig. 18 is acquired by finding the result with the smallest smoothing error under the corresponding noise level. The estimations of s by GCV are also marked in the figure for comparison. It shows that GCV method always choose a decent smoothing parameter, which is very close to the best DFS results at different noise levels. In fact, the differences between the smoothing errors resulting from GCV estimation and the ones using the best s are always less than 0.1 %. It suggests that the GCV method is a good adaptive method for different noise levels. Another interesting result is that the optimal smoothing parameter s is proportional to the noise level. Therefore, it seems possible to estimate the noise level of the experimental data according to the GCV estimation of smoothing parameter s.

Besides the theoretical proof (Craven and Wahba 1978) and the above numerical test, the GCV method, as a general method, has been successfully applied in the DCT-PLS method on dealing with various types of experimental errors (Garcia 2010). In current work, we also perform a systematical test of DFS on all possible errors of PIV data in the following Sects. 3–5. The good performances of both DCT-PLS and DFS in practical application strongly evidence that the GCV method is an effective and robust algorithm in choosing the smoothing parameter.

At last, it is worth to note that any smoothing operation has certain low-pass filtering effect for the flow field. For DFS, larger smoothing parameter means stronger noise-reduce effect and, therefore, larger range of frequency truncation. DFS seeks a good balance between the two effects of noise reduction and spatial attenuation by using the GCV method.

Fig. 18

Test of GCV method. The contours indicate the relative smoothing error. Two curves with different markers stand for the best parameter s and the GCV-estimated s