A direct measure of the frequency response of hot-wire anemometers: temporal resolution issues in wall-bounded turbulence

Abstract

Based on the need to characterise the accuracy of hot-wire anemometry (HWA) in high Reynolds number wall-bounded turbulence, we here propose a novel direct method for testing the frequency response of various systems to very high frequency velocity fluctuations (up to 50 kHz). A fully developed turbulent pipe flow is exploited to provide the input velocity perturbations. Utilising the unique capabilities of the Princeton Superpipe, it is possible to explore a variety of turbulent pipe flows at matched Reynolds numbers, but with turbulent energy in different frequency ranges. Assuming Reynolds number similarity, any differences between the appropriately scaled energy spectra for these flows should be indicative of measurement error. Having established the accuracy of this testing procedure, the response of several anemometer and probe combinations is tested. While these tests do not provide a direct or definitive comparison between different anemometers (owing to non-optimal tuning in each case), they do provide useful examples of potential frequency responses that could be encountered in HWA experiments. These results are subsequently used to predict error arising from HWA response for measurements in wall-bounded turbulent flows. For current technology, based on the results obtained here, the frequency response of under- or over-damped HWA systems can only be considered approximately flat up to 5–7 kHz. For flows with substantial turbulent energy in frequencies above this range, errors in measured turbulence quantities due to temporal resolution are increasingly likely.

Appendices

Appendix 1: Variation of response within a boundary layer

Figure 11a shows the square-wave response measured in the freestream of a boundary layer wind tunnel (the Melbourne HRNBLWT) at various mean velocities ${\overline{U}} = 10$, 3 and $1\,\hbox {ms}^{-1}$ (black, red and blue curves, respectively). The HWA system consists of CTA1 and a standard 55P15 probe. The anemometer has not been adjusted between these experiments, and thus, any difference between the three curves in Fig. 11 is purely a result of the altered mean velocity. Figure 11b shows the corresponding normalised energy spectrum of the square-wave response, which for CTA1 is shown to give a reasonable representation of the directly determined response $\chi$ (see Fig. 3). The variation in mean velocity over the sensor causes a pronounced change in the measured response, shifting from an under-damped system at $10\,\hbox {ms}^{-1}$ to a clear over-damped behaviour at 1 ms$^{-1}$. Such behaviour is to be expected from the heat transfer equation. As ${\overline{U}}$ reduces from $10$ to $1\,\hbox {ms}^{-1}$, the wire Reynolds number has been decreased by a factor of approximately 10 reducing the heat transfer or cooling from the wire by a factor of approximately 3. Reduced heat transfer leads to increased damping. This is evident in Fig. 11 with the system switching from under- to over-damped behaviour as the mean velocity reduces.

Fig. 11

Change with local mean velocity of a the square-wave response and b the normalised energy spectrum of the square-wave response. Black ${\overline{U}} = 10.7\,\hbox {ms}^{-1}$, red ${\overline{U}}= 3.5\,\hbox {ms}^{-1}$, blue ${\overline{U}} = 1.0\,\hbox {ms}^{-1}$. Tuning of anemometer has not been altered (changes are due only to the change in ${\overline{U}}$)

Figure 11 suggests that it is essential to check the square-wave response across the entire expected velocity range prior to a boundary layer traverse experiment. If the frequency response is only checked at freestream conditions (say $10\,\hbox {ms}^{-1}$), a system could encounter substantial temporal resolution issues close to the wall as the system becomes increasingly damped. For a wall-bounded turbulent flow with a freestream velocity of $10\,\hbox {ms}^{-1}$, local mean velocities of $3$ and $1\,\hbox {ms}^{-1}$ would be encountered at $z^{+} \approx 10$ and 3. Hence, the temporal response of the system would shift from under-damped to over-damped as the probe was traversed very close to the surface.

Appendix 2: Determining the square-wave response

The square-wave responses shown throughout this investigation were obtained with the probe at the pipe centreline, with the centreline velocity and pressurisation matched to the local conditions at the probe location (which was either $z^{+} = 29$ or 80) for the baseline experiment $e5$. Since the pipe core is fully turbulent, the injected square-wave response is superimposed over a turbulent signal. For analysis, the signal is sampled at 1.2 MHz for a duration of 10 s. An example of this sampled signal for the under-damped CTA1 (as detailed in Figs. 1, 2, 3) is shown in Fig. 12a, with an inset showing an enhanced detail of the square-wave response superimposed onto the turbulent core flow. To separate the square-wave response from the turbulent signal, the sampled data are filtered using a sharp spectral high-pass filter at $f = 7$ kHz. An example of the resulting filtered signal is shown in Fig. 12b. Note that the cut-off filter was selected to remove the large-scale turbulent component from the signal. A simple threshold technique is used to extract the square-wave response. In the case shown, a threshold of 0.04 V was used detect the rising edge of the square-wave input. Regions where the rising part of the signal exceeded this threshold are shown by the dot symbols on Fig. 12b. A region of the signal surrounding this threshold crossing is then extracted and added to an ensemble average as shown in Fig. 12c. The grey curves show the square-wave responses extracted from Fig. 12b. The black curve shows the average response.

Fig. 12

Plots illustrating the extraction of the square-wave response from a turbulent signal measured in the core. a Original time series of the velocity output from the anemometer (sampled at 1.2 MHz) showing square-wave response superimposed onto turbulent signal, inset shows an expanded detail of the response to rising step function. b Data high-pass filtered at 7 kHz, dashed line shows detection threshold and red dots show rising edge detection. c Ensemble average of detected rising edge step responses, grey lines shows the individual detections shown in plots a and b and black curve shows ensemble average over 10 s of data (approximately 10,000 individual occurrences)