A scheme to correct the influence of calibration misalignment for cross-wire probes in turbulent shear flows

Abstract

Velocity fluctuations, measured via multi-wire probes, are very sensitive to misalignment between the calibration coordinate system and that of the wind tunnel. The present study proposes a scheme to correct the erroneous velocity fluctuations processed from a misaligned calibration while investigating a wall-bounded turbulent shear flow. The scheme is based on the premise that the viscous-scaled spectral energy distribution in the small-scales is invariant with Reynolds number and solely depends on the viscous-scaled spatial resolution of the sensor. Energy spectra processed from the misaligned calibration, in this small-scale range, are compared with the ‘expected’ spectra obtained via synthetic experiments on a direct numerical simulation data set. The erroneous lateral velocity spectra is found to be either relatively amplified or attenuated, by almost the same factor, at all wall-normal distances across the shear flow. A unique gain, defined to be the correction ratio, is thus obtained by forcing the erroneous spectra onto the reference spectra in this scale range. This ratio is further used to rectify the time series of the lateral velocity fluctuations, acquired across the shear flow, via Fourier analysis. The scheme is shown to be effective for experiments conducted across a decade of Reynolds number and using probes of varying spatial resolution.

Graphic abstract

Appendices

Appendix 1: Accounting for the drift in hotwire voltage

It is a very well known fact that hotwire sensors drift owing to various reasons (Talluru et al. 2014) and this phenomena can lead to erroneous velocity statistics unless accounted for. In this regard, we have implemented the practice of doing periodical ‘free-stream’ checks (Talluru et al. 2014) for the X-probe to obtain an updated 1-D calibration curve (for both sensors) at each wall-normal location of the profile (refer Sect. 2.1.3). These 1-D calibration curves are used as references to generate unique u and v (or w) calibration surfaces at corresponding $z^{+}$. To this end, we process the 2-D calibrations via the voltage offset (VO) method demonstrated previously (Sect. 3.1) and show that the sensor drift is implicitly accounted on using this method. Through this, we intend to strengthen our argument that the attenuated lateral velocity statistics (shown in Figs. 3b, 4b), obtained on processing the data via the misaligned calibration, are an artifact of the misalignment and not the hotwire sensor drift.

Fig. 12

Viscous-scaled profiles of a mean streamwise velocity and streamwise turbulence intensity and b spanwise turbulence intensity obtained on processing the high $\hbox {Re}_{\tau }$ boundary layer data set using the accurate and drifted 2-D calibrations. Data processing, in case of the drifted calibration, is carried out using the VO method

To demonstrate this, an additional drifted 2-D calibration, apart from the ‘accurate’ and misaligned 2-D calibrations, was performed on the same uv-CX probe used in the $\hbox {Re}_{\tau }$ ${\approx }$ 10,000 experiments in the ZPG TBL (Table 1). It was performed approximately 12 h after the end of measurement to ensure that the sensor drifts sufficiently. During this calibration, the jet coordinate system was aligned with the measurement coordinate system in the same manner as done for the accurate 2-D calibration, to rule out any misalignment. Figure 11a shows the raw voltage pairs acquired during the accurate and the drifted 2-D calibration on the uv X-probe. Also plotted are the mean voltages acquired during the 1-D calibration done immediately before the measurements. Similar to Figs. 2b and 11b shows the raw calibration voltages fitted to smooth functions of jet velocity and angles based on the EAM and solved for a linearly distributed set of $U_{\mathrm{jet}}$ and $\theta$ for a meaningful comparison. The difference between the drifted and the accurate 2-D calibration is pretty clear. The voltage drift is apparent from the difference between the voltage pairs from the two calibrations at $U_{\mathrm{jet}}$ ${\approx }$ 0. It is different from the case of a misaligned 2-D calibration (Fig. 2b), where a difference was noted only for voltages acquired at $U_{\mathrm{jet}}$ > 0.

Following the VO method, the voltages corresponding to ${\theta }_{o}$ for the drifted 2-D calibration are forced to be equivalent to the 1-D calibration voltages, through which a set of unique voltage offsets (${\Delta}E_{1}$($U_{\mathrm{jet}}$), ${\Delta}E_{2}$($U_{\mathrm{jet}}$)) is estimated for both the sensors. This offset is applied across the entire calibration map, resulting in a ‘voltage offset’ 2-D calibration map shown in Fig. 11c. On performing the voltage shift, the difference between the accurate and drifted 2-D calibration appears to be negligible, suggesting that the effect of drift has been accounted by processing through the VO method. To confirm this, we process the boundary layer dataset acquired with the same probe via the two calibration surfaces. A good agreement is observed between the mean statistics (Fig. 12) processed via the two calibrations.

Appendix 2: Methodology to correct the time series of v or w

Let us consider the time series of the uncorrected lateral velocity fluctuations as q(m), where q represents v or w and m = 1, 2 $\ldots N$ represents the discrete samples of the fluctuations with N equaling the total number of samples (sampling frequency times the total time period of acquisition). Similarly, let the corrected time series be denoted as ${{q}^{\mathrm{c}}}(m)$. If ${\widetilde{Q}}(n)$ and ${\widetilde{Q}^{\mathrm{c}}}(n)$ denote the complex Fourier coefficients obtained on computing the Fourier transform ($\mathcal {F}$) of q(m) and ${q^{\mathrm{c}}}$(m), respectively, then ${{\widetilde{{Q}^c}}}(n)$ = ${\sqrt{r^{\mathrm{c}}}}{{\widetilde{{Q}}}}(n)$ according to the correction scheme proposed in Sect. 4.2, with n being the mode number. Here, ${\sqrt{r^{\mathrm{c}}}}$ being real-valued will only influence the magnitude of the Fourier coefficient. The time series of the corrected velocity fluctuations can thus be found by simply computing the inverse discrete Fourier transform for ${{\widetilde{{Q}^{\mathrm{c}}}}}(n)$:

$$\begin{aligned} \begin{aligned}&{q^{\mathrm{c}}}(m) = {{\mathcal {F}}^{-1}}[{\widetilde{Q^{\mathrm{c}}}(n)}] = \frac{1}{N} {\sum _{n=0}^{N-1} {{{\widetilde{Q}}^{\mathrm{c}}}(n)} \hbox {exp}\left( \frac{i2{\pi }mn}{N} \right) }, \hbox {or}\\&{q^{\mathrm{c}}}(m) = \frac{1}{N} {\sum _{n=0}^{N-1} {\sqrt{r^{\mathrm{c}}}}{{\widetilde{Q}}_{n}} \hbox {exp}\left( \frac{i2{\pi }mn}{N} \right) } \end{aligned} \end{aligned}$$

(3)

Fig. 13

a, c Corrected viscous-scaled wall-normal turbulence intensity profiles and b, d premultiplied energy spectra of the wall-normal velocity at $z^{+} {\approx }$ 100 (marked by black line in a, c) obtained from various X-probes for a channel flow. These profiles are selectively plotted in a way to demonstrate the effect of varying ${{\Delta}{s^{+}}}$ (a, b) and $l^{+}$ (c, d). Color coding corresponding to the X-probes is given in Table 1. Dark and light shading represents data from the $\hbox {Re}_{\tau } \approx$ 1000 channel flow experiments and the corresponding synthetic experiments, respectively. Note the vertical shift in profiles for the synthetic experiments

Appendix 3: Investigating the effect of X-probe spatial resolution

The three types of X-probes chosen to conduct experiments in the channel flow have systematically varying $l^{+}$ and ${\Delta}{s^{+}}$ (refer Table 1). Here, we compare the trends observed in the corrected experimental dataset, due to variation of these parameters, with those from the corresponding synthetic experiments. For brevity, we restrict ourselves solely to studying these for the uw X-probe.

Figure 13 shows the corrected ${\overline{w^2}}^{+}$ and ${{{k}^{+}_{x}}{{\phi }^{+}_{ww}}}$ at $z^{+}$ $\approx$ 100 from the three different uw X-probes. The ${\Delta}{{s^{+}}}$ and $l^{+}$ trends observed in the ${\overline{w^2}}^{+}$ profiles from the corrected experimental data are consistent with those from the synthetic experiments. It is interesting to see the qualitative agreement between the pair of spectra, from the two sources, in Fig. 13b, d. For the large scales (${{\lambda }^{+}_{x}}>$ 300), the difference in the energy distribution for varying ${\Delta}{{s^{+}}}$ is very similar in both the datasets. Similarly, for the case of varying $l^{+}$, the spectra from the corrected experimental dataset nearly overlap in the large-scale range as seen for the synthetic experiment dataset. The consistency of the corrected dataset with the synthetic experiments demonstrates the effectiveness of the correction scheme, which is facilitated by the availability of DNS flow fields to simulate the ‘expected’ spectra in the small-scale range.

Appendix 4: Database of energy spectra obtained from synthetic experiments

A database of the viscous-scaled premultiplied energy spectra, obtained via synthetic experiments for varying measuring volumes of the X-probes, can be accessed at http://fluids.eng.unimelb.edu.au/. The users can follow the steps summarized in Sect. 6 to correct their X-probe dataset for canonical wall-bounded flows in case of calibration misalignments.