Denoising of time-resolved PIV for accurate measurement of turbulence spectra and reduced error in derivatives

Abstract

The accuracy of time-resolved PIV (TRPIV) turbulence measurements is limited by white noise, which reduces the signal-to-noise ratio (SNR) of small-scale velocity fluctuations. This paper demonstrates a novel energy filter that extends the concept of spectral noise subtraction to the time domain. The filter is equivalent to the spectral subtraction of white noise energy, and therefore it can recover the true signal energy. Its effectiveness is evaluated by comparing two-component (2C) TRPIV and constant temperature anemometry (CTA) measurements performed in grid-generated turbulence (Re _λ = 90). The denoised PIV measurements exhibit a SNR equivalent to those of a high-performance CTA system. The temporal spectra of velocity fluctuations and derivatives are accurately recovered, with an improvement in dynamic range by a factor of $\fancyscript{O}(10^3)$. The error in dissipation estimates derived from the frequency spectrum is reduced to approximately 2 %. The correlation coefficient between spatial gradients computed directly and those computed via Taylor’s hypothesis improves from 0.83 to 0.95. The mean-square error reduction is found to be equivalent in the frequency and wavenumber domains, although we observe that frequency domain filtering has a limited ability to improve the SNR of spatial spectra at high wavenumbers. The performance of the energy filter is shown to be sensitive to the convergence of the measured spectra; therefore, we provide sampling criteria to ensure optimal implementation in measurements of turbulent flows.

Appendix: Comparison between the energy and the Wiener filter

This appendix compares the energy filter to the white noise form of the Wiener filter used in the denoising assessment by Vétel et al. (2011). Although these filters bear some resemblance, there are crucial differences that motivate the choice of the energy filter for denoising measurements of turbulence. We show that while the Wiener filter is optimal in the sense of error minimisation, it leads to a bias of the energy spectral density.

Using our notation, the transfer function of the Wiener filter is (see Press et al. (1988) for details):

$$ H^{\rm W} = \left[1-\frac{\phi_{\varepsilon_i}}{\hat{\phi}_{ii}}\right]^2. $$

(25)

which is simply the square of the energy filter’s transfer function, H. H ^W is obtained by minimising the mean-square error between the true and denoised signals:

$$ \langle | \tilde{u}(t)-u(t) |^2 \rangle = \int_{-\infty}^{+\infty} |\tilde{X}_f - X_f |^2 \text{d}f. $$

(26)

This minimisation might be expected to produce the same result as the energy filter (which is defined to be equivalent to white noise subtraction) however it does not. To illustrate why, the integrand of Eq. 26 can be expanded as:

$$ |\tilde{X}_f - X_f |^2 = |\tilde{X}_f|^2 + |X_f|^2 - |\tilde{X}_f \cdot X_f^* + \tilde{X}_f^* \cdot X_f|, $$

(27)

where (*) denotes the complex conjugate. The expansion reveals that the minimisation problem accounts for both magnitude and phase errors. However, this is a constant phase filter that can only affect the signal magnitudes, $|\tilde{X}_f|$. The two filters produce different results because the Wiener filter compromises between magnitude and phase errors when minimising Eq. 27, whereas the energy filter minimises the magnitude error only. At high frequencies, where the SNR is poor, this compromise is manifested as a negative bias of energy.

To demonstrate this, we generated a synthetic signal, u(t), with a turbulence-like spectrum $\phi(f)=f^{-5/3} \cdot e^{-a\cdot f}$. Gaussian white noise was added in the time domain to create the noisy signal, $\hat{u} (t)$. We arbitrarily chose to use Gaussian distributions; however, any other distributions could have been used as long as the signal and the noise have zero correlation. The parameter ‘a’ and the noise magnitude were chosen arbitrarily to achieve a SNR equivalent to the raw PIV data, that is, $\langle \varepsilon_u^2 \rangle / \langle u^2 \rangle =0.05$.

The spectra of the true, noisy, and denoised signals are compared in Fig. 17. The results confirm that the energy filter recovers the true spectrum, whereas the Wiener filter induces a significant negative bias at high frequencies. It also creates a positive bias at low frequency that is not visible here due to the logarithmic axes. Note that this observed bias is consistent with the denoised spectra of real turbulence data in Fig. 12 of the study by Vétel et al. (2011).

Fig. 17

Comparison of the denoising performance of the energy and Wiener filters applied to synthetic data: open diamond true spectrum ϕ, red lines ${\phi}$ true spectrum with added white noise $\hat{\phi},$ blue lines denoised by the Wiener filter $\tilde{\phi},$  green lines denoised by the proposed energy filter $\tilde{\phi}$

The relative error in the standard deviation, mean-square velocity derivative, and correlation coefficient with the true signal (three statistical quantities which are of practical interest for turbulence measurements) are compared in Table 10. It can be seen that the energy filter provides the best estimate of quantities related to the energy spectrum, whereas for the correlation coefficient, the Wiener filter performs slightly better. In summary, the Wiener filter yields the optimal correlation between the true and denoised signals (although in practice the difference is not significant), but does so at the expense of a biased energy spectrum.

Table 10 Comparison of synthetic data statistics. All units are [%] error