Comparative Testing the Uncertainty of A/D Converters in A Hot-Wire Anemometer Measurement Using Grid-Generated Turbulence

This study investigates the uncertainty of the A/D converter when measuring multi-scale turbulence fluctuations using a hot-wire anemometer. The flow is generated in a low-speed wind tunnel, the turbulence is generated by a turbulence grid and the fluctuations are measured. A compact hot-wire anemometer was used and a high accuracy A/D converter was introduced for the measurements. Methods to minimise the influence of noise were also incorporated and measurements were carried out in two situations, one with and one without turbulence. The experimental results showed that measurements under a specific condition (Case 1 in this study) had the sufficienty reproducibility of turbulence fluctuations and the least effect of noise. The results showed that the high frequency characteristics of the turbulence could also be accurately measured under this condition. In conclusion, it is clear that the choice of a suitable A/D converter and the optimisation of the measurement environment are critical for turbulence measurements with a hot-wire anemometer.


Introduction
In order to improve the accuracy of numerical analysis of turbulence, models and schemes have been developed and improved.A large amount of data has been obtained from experiments simulating real conditions in all types of fluids.Wind tunnel experiments on incompressible fluids are one of these and are considered to be one of the most widely used methods for understanding the characteristics of turbulent flows [1].In these wind tunnel experiments, turbulence is introduced into the flow field by means of a turbulence-generating grid [2].In recent years, multi-scale turbulence generating grids have received increasing attention [3][4][5][6][7][8][9][10]. It has been suggested that the multiscale turbulence generated by these grids is higher in turbulence than conventional turbulence and that the dissipation characteristics can more accurately reproduce realistic phenomena, and therefore more experimental data is required.Among the many measurement methods, hot-wire anemometry [11][12][13][14][15][16][17] is expected to be able to measure smaller fluctuations than a particle image velocimetry or ultrasonic anemometry.
The turbulence fluctuations are expected to have a certain degree of universality on small scales when the Reynolds number is high.To measure these small-scale fluctuations experimentally, measurement techniques with high temporal and spatial resolution are required.Furthermore, the amplitude of these turbulent fluctuations is so small that very low noise is required to measure them.By further increasing the turbulence intensity, turbulent fluctuations are present at higher frequencies.Therefore, lower noise at high frequencies is required to measure these high turbulence fluctuations at higher turbulence intensities.Hot-wire anemometers are characterized by high temporal resolution and low noise [18][19][20].One of the high frequency noise components of hot-wire anemometers is electrical noise from an A/D converter.This noise is caused by the bit depth of the A/D conversion and the noise in the circuits of a converter.
The hot-wire anemometer measurement can therefore capture minute fluctuations by preventing unnecessary signals from being introduced into the measurement as a result of circuit miniaturization.When analyzing the small fluctuation data obtained in this way, it is critical that the intensity of the noise in the data is large.There are a few factors that can lead to the inclusion of non-fluctuation noise signals in the measurement results.These include electrical factors such as PCs and magnetic fields, and in the case of AC power supplies, factors such as changes due to increased use.This study focuses on the noise introduced during the conversion and recording of signals from hot-wire anemometers.A/D converters are used for logging and conversion, but there are few examples showing the effect of differences in performance, such as the number of bits, on the accuracy of turbulence measurements.This study attempts to investigate the factors responsible for this effect, which could lead to an improvement in the measurement uncertainty of the overall hot-wire velocity measurement system.
The aim of this study is to investigate the uncertainty between A/D converters in the measurement of multi-scale turbulence fluctuations using a hot-wire anemometer.A low-speed wind tunnel is used to reproduce the flow field in a test section where turbulence is introduced by a turbulence grid.Turbulence fluctuations are measured using a miniaturized hot-wire anemometer.Experiments are carried out at different flow velocities to investigate and validate the flow and non-flow conditions.The effect of the grounding on the measurement uncertainty is also tested.The differences in turbulence statistics and higher order statistics are calculated for the time series data obtained and used to determine the turbulence characteristics.

Experimental Methods
Low-speed wind tunnels are often used to generate grid turbulence fields.The experimental equipment used to generate the flow field is described.A flow field is generated in the test section, which is the test section, by the low-speed wind tunnel apparatus at Okayama University [21].The cross-sectional area of the low-speed wind tunnel outlet is the same as the cross-sectional area of the test section, which is a square with a length of 0.4 m per side.The total length of the test section is 5.0 m.A turbulence grid is installed at the entrance of the test section to introduce turbulence into the flow field.The measurement coordinate system takes the grid centre as the origin, with the x, y and z directions being the flow direction, the vertical direction and the span direction respectively.Measurements are made by traversing the hot-wire anemometer in the flow direction.Previous studies have shown that the effect of freestream acceleration due to wind tunnel blockage on turbulence in this wind tunnel test section is not significant [22,23].
The turbulence grid installed at the entrance to the test section is described.A fractal turbulence grid was used to introduce the turbulence.The grid has a self-similar shape and the components of this grid are square grids, which are often used in previous studies.The blockage factor of this grid is 0.32.The fractal dimension of this grid is 2. The number of iterations of this grid is 3. Previous studies have shown that this fractal turbulence grid can induce stronger turbulence than a square grid.In addition, the previous studies have shown that this grid can efficiently generate multi-scale generated turbulence.In the present study, the characteristic length of this flow is taken to be the length of one side of the channel cross-section of the test section, denoted by D.
The hot-wire anemometer used to measure fluctuations in the generated multi-scale turbulent flow field is described.A constant-temperature hot-wire anemometer was used in this study [24,25].This anemometer is compact in size so that it can be installed in the flow channel.An I-type hot-wire probe is installed at the leading edge of the hot-wire anemometer.The material of the hot wire is tungsten and the wire diameter is 5 µm.The flow fluctuations obtained by the hot-wire anemometer are output as a voltage value.This output value must be obtained using a data logger and converted to a digital signal.The resolution is 24 bits and the sampling rate is up to 256 kHz.The maximum number of channels is 2, which is supplied with the main unit.Logging is performed by connecting both transducers to a PC and using dedicated software to set the trigger conditions.Experiments were carried out with noise covers on the input cables of the turbulence signals to the transducers.Unused channels were also grounded to remove unknown noise.[kHz] from the square wave verification test, so the sampling frequency was set to 50 [kHz] based on this.For Case 0, the sampling was carried out on Channel 2, which has the lowest noise level of all eight channels; for Case 1, the measurement was carried out on Channel 1.As in Case 0, a comparison test was carried out with a ground wire connected to Channel 2 as in Case 1 and without a ground wire as in Case 2.

Results
The voltage data of the variations obtained by the hot-wire anemometer at the measurement point x/D = 10 under the condition Uo = 0 [m/s] are shown in figure 2. The vertical axis of the graph shows the output voltage normalised to the mean voltage and the horizontal axis is the measurement time.The black plots in the diagram show the results of Case 0 and the values obtained by the Keyence data logger, while the red plots in the diagram show the results of Case 1 and the values obtained by the Turtle Industries A/D converter.As can be seen from the figure, the upper limit of the amplitude variation of the voltage values in Case 0 reaches about 0.3%, whereas the variation of the amplitude of the voltage values in Case 1 is less than 0.05%, which is sufficiently small compared to Case 0. The measurement results under these conditions can be regarded as synonymous with the noise reflected in the output results of the hot-wire measurement, so that the measurement uncertainty in Case 1 can be regarded as smaller than in Case 0. This result suggests that the increased number of bits in Case 1 compared to Case 0 acts to reduce the noise in the measurement, resulting in better measurement results.3, the scatter of the voltage data profiles is smaller in case 1 than in case 0. Focusing on this signal, the multi-scale generated turbulence is constant downstream of x/D = 10.The amplitude of the fluctuations is also small, around 1% of the mean flow velocity, indicating that the inhomogeneity of the turbulence is also sufficiently small downstream of x/D = 10.Furthermore, from the waveforms of the time series data in both figures, the fluctuations appear to be typical turbulence signals, such as lattice turbulence, so that the measurement of the fluctuations using the hot-wire anemometer can be considered to have been carried out satisfactorily.The voltage data of the multi-scale turbulence fluctuations in Case 0 and Case 1 obtained in figure 2 and figure 3 are quantitatively evaluated.Turbulence statistics are calculated and compared as an indicator for the evaluation of the low frequency band in subsequent measurements.The relative RMS is used as a typical turbulence statistic, e21/2/E, where e is the measured voltage variation and E is the mean voltage.Also,   denote the ensemble average.Skewness and flatness factors are then used.If a fluctuation follows the Gaussian distribution, the values of the measured data correspond to 0 and 3 respectively.
The calculated results of the turbulence statistics are shown in figure 4.  In order to evaluate the spectral characteristics of both transducers in the high frequency range, power spectra are calculated.The measurements were carried out at Uo = 12 [m/s] under conditions where turbulence was introduced.Figure 5(a) shows a linear presentation of the calculated power spectrum.The black and red solid lines show the results for Case 0 and Case 1 respectively, while the blue solid line shows the results for Case 2. The longitudinal axis of figure 5(a) shows the power spectrum normalised by the square of the mean voltage, while the lateral axis is the wavenumber normalised by the Nyquist wavenumber.As shown in the figure, the values of Case 0 are larger than those of Case 1 and Case 2 at high frequencies.This suggests that the noise present in Case 0 is reduced in Cases 1 and 2. Grounding Channel 2 also removes noise in certain frequency ranges.Figure 5(b) shows the calculated results of the power spectrum in one logarithm to observe in detail the characteristics observed in figure 5(a).As in figure 5(a), the solid black line shows the results for Case 0, the solid red line for Case 1 and the solid blue line for Case 2. As in the linear display, the measurement results for Case 0 contain significant noise in the high frequency range and the output values are larger than the other two results, with output values approximately 100 times higher.In contrast, the output results of Case 1 do not contain the noise seen in Case 0, and the noise seen in the specific wavenumber band in the output results of Case 2 is as high or higher than the noise in Case 0. Therefore, it is necessary to ground Channel 2 as in Case 1.
To further clarify the differences between Case 0 and Case 1 in this high frequency region, the bispectrum is examined by calculating the triple correlation function of the voltage fluctuations obtained from the measurements [26][27][28].The details of this method of analysing the higher order statistics of turbulence using the bispectrum are described in detail in the literature on CTA measurements focusing on frequency response.The calculation of the bispectrum is described below.The triple correlation of a voltage fluctuation is related to the convection term in the governing flow equations: e(t)2 e(t + t) / e(t)23/2.Here t and t denote the distance between two points and the time, respectively.Note that the triple correlation is normalised to the RMS value of the voltage variation.The bispectrum is obtained by Fourier transforming the normalised triple correlation.
The calculated results of the bispectrum derived from the triple correlation of the voltage fluctuations are shown in figure 6.As shown in the figure, the distribution of the voltage variation bispectrum decreases with increasing frequency.The order of the bispectrum in Case 0 is approximately 102 times larger than that in Case 1, which is in general agreement with the calculated results of the power spectrum.The increase in the bispectrum value seen in figure 5 for k/kN = 0.8 ~ 0.9 in the calculated results for Case 1 is also evident.In addition, focusing on the effective range of the spectral characteristics of turbulence, it can be seen that k/kN = 0.2 is at most k/kN = 0.2 in Case 0 while k/kN = 0.6 in Case 1.This difference should not be underestimated as the signals extend into the high frequency range as the turbulence intensity increases.

Discussion
In this study, the relative RMS value is less than 0.0005, as shown in figure 4(a), indicating that the measurements were made for weak turbulent fluctuations.The case of a flow field with stronger turbulent fluctuations is considered.In this case, the relative RMS values are assumed to be approximately a few percent of the free stream.Figure 5 shows that for weak turbulence fluctuations, the turbulence signal appears only in the range of k/kN = 0.2 ~ 0.4.In the case of strong turbulence fluctuations, the turbulence signal is expected to have a larger power spectrum and to appear in the higher frequency region.Under the measurement conditions of Case 0, the spectrum formed in the high frequency region contains noise of the order of 10 -7 .In contrast, under the measurement conditions of Case 1, at least in the region of k/kN = 0.8, the noise seen in Case 0 can no longer be included.From the above, the results of this study are expected to be highly relevant to the measurement of stronger turbulent fluctuations.
The response frequency of the instrument, which is treated as the measurement resolution when measuring turbulent fluctuations, is discussed.The response frequency of a hot-wire anemometer is considered to be several tens of kHz, and the signals measured with this resolution are obtained as data by the A/D converter used in this study to check whether the data are covered.Figure 5 shows the horizontal axis normalised to the Nyquist frequency of the response frequency, and the signals indicating turbulence fluctuations are at most up to k/kN = 0.4.Beyond this point, there are no signals to indicate turbulence fluctuations.Therefore, this measurement can be recorded with sufficient resolution as an A/D converter for turbulence fluctuation measurement using a hot-wire measurement system and is considered suitable for fluctuation measurement.

Conclusion
The aim of this study is to investigate the uncertainty of the A/D converters used in the hot-wire measurement of grid turbulence.Two transducers were used for the verification, each measured with the same hot-wire anemometer.The measurements were made using a low-speed wind tunnel at Okayama University and a fractal turbulence grid to reproduce the multi-scale turbulence in the test section, and the fluctuations in turbulence were measured as voltage values using an I-type hot-wire probe and a constant-temperature hot-wire anemometer.The time series data of the voltage fluctuations were compared and then the differences in the output data between the two converters were compared.The relative RMS values, skewness and flatness were calculated from the time series data of the voltage From the time series data of the fluctuations, the output results in Case 1 were found to be smaller in the case of no flow.Furthermore, the calculated turbulence statistics such as relative RMS, skewness and flatness showed the measurement reproducibility of the turbulence fluctuations by both converters and that the distribution of the multi-scale turbulence in the downstream was close to Gaussian.The calculation of the power spectrum of the voltage fluctuations showed that case 1 was least affected by noise, indicating the need to ground another channel.Furthermore, the calculated bispectrum of the fluctuations showed that Case 1 reduced the output result in the high frequency range to about 1/100 of that of Case 0. These results show that the measurement in Case 1 is capable of reducing the measurement uncertainty in the present measurement environment and that the measurement of turbulence characteristics over a high frequency range is achievable.

Figures 1 (
Figures 1(a) and 1(b) show devices used to acquire and record the voltage data of the fluctuations obtained by the hot-wire anemometer.Figure 1(a) shows an A/D converter manufactured by Keyence.The bit depth is 16.The maximum number of channels is 8.The channels are separated from the logger and the signal is carried by a cable.Figure 1(b) shows an A/D converter manufactured by Turtle Industry.The resolution is 24 bits and the sampling rate is up to 256 kHz.The maximum number of channels is 2, which is supplied with the main unit.Logging is performed by connecting both transducers to a PC and using dedicated software to set the trigger conditions.Experiments were carried out with noise covers on the input cables of the turbulence signals to the transducers.Unused channels were also grounded to remove unknown noise.

Figure 1 .
Figure 1.Schematic of the two A/D converters tested in this experiment.(a) Keyence NR-600 (Case 0), (b) Turtle Industry TUSB-0224ADM (Case 1).In this study, to investigate the conversion uncertainty between the two A/D converters mentioned above and the noise level with and without grounding, the flow fluctuations are measured and the voltage data are compared.An analogue oscilloscope was used for these measurements and the voltage signals were checked.The test conditions are described below.The flow velocity conditions set were Uo = 0 [m/s] and Uo = 12 [m/s].The sampling time was 60 [s].The cut-off frequency fc was approximately 28.6 [kHz] from the square wave verification test, so the sampling frequency was set to 50 [kHz] based on this.For Case 0, the sampling was carried out on Channel 2, which has the lowest noise level of all eight channels; for Case 1, the measurement was carried out on Channel 1.As in Case 0, a comparison test was carried out with a ground wire connected to Channel 2 as in Case 1 and without a ground wire as in Case 2.

1 Figure 2 .
Figure 2. Voltage time series signals normalised to the mean voltage at zero flow velocity in each hotwire anemometer.The voltage data of the fluctuations obtained at x/D = 10 measurement points are shown in figure 3(a) and figure 3(b), respectively, under the conditions of multi-scale generated turbulence with Uo = 12 [m/s] in the test section.As in figure 2, the x-axis is the output voltage normalised to the mean voltage and the y-axis is the measurement time.Comparing the waveforms in (a) and (b) of figure3, the scatter of the voltage data profiles is smaller in case 1 than in case 0. Focusing on this signal, the multi-scale generated turbulence is constant downstream of x/D = 10.The amplitude of the fluctuations is also small, around 1% of the mean flow velocity, indicating that the inhomogeneity of the turbulence is also sufficiently small downstream of x/D = 10.Furthermore, from the waveforms of the time series data in both figures, the fluctuations appear to be typical turbulence signals, such as lattice turbulence, so that the measurement of the fluctuations using the hot-wire anemometer can be considered to have been carried out satisfactorily.

Figure 3 .
Figure 3.Comparison of a relative instantaneous voltage variation between two A/D converters corresponding to the relative instantaneous velocity signal of the decaying turbulence in the downstream.

Figure 4 (
a) shows the calculated relative RMS values of the voltage fluctuations.As shown in the figure, the relative RMS values are approximately the same.The values are less than 0.005, indicating that the residual turbulence fluctuations are less than 0.5% of the main flow velocity.Figures 4(b) and 4(c) show the calculated results of the skewness and flatness of the voltage fluctuations.As shown in the figures, the skewness and flatness values are of similar magnitude between Case 0 and Case 1.The skewness and flatness are close to 0 and 3 respectively, indicating that the multi-scale generated turbulence follows a Gaussian distribution at the x/D = 10 measurement point.The calculation results for these three indices validate the reproducibility of the multi-scale generated turbulence measurement in this study.

Figure 4 .
Figure 4. Comparison of the relative RMS voltage fluctuation and the skewness and flatness of the voltage fluctuations between the two A/D converters.

Figure 5 .
Figure 5.Comparison of the normalised power spectrum among the three conditions.Here Case 2 is the case without grounding on another BNC connector in Case 1.

Figure 6 .
Figure 6.Comparison of the normalised bispectrum of the voltage signal obtained from the triple correlation function between two A/D converters.
Conference on Mechanical Engineering and Materials Journal of Physics: Conference Series 2694 (2024) 012003 IOP Publishing doi:10.1088/1742-6596/2694/1/0120038fluctuations as typical statistics of turbulence.The power spectrum of the voltage fluctuations was calculated to check the frequency characteristics.The frequency response was also examined using the bispectrum of the voltage fluctuations.