Experimental validation on a calibration position of a hot-wire anemometer for measuring multi-scale grid-generated turbulence

This study aims to experimentally examine conditions that should be satisfied by a calibration position of a constant-temperature hot-wire anemometer. Specifically, this study addresses an appropriate measurement point when this calibration is performed downstream of a turbulent field generated by a multi-scale turbulence grid. The hot-wire measurement in this study is based on an I-type probe with a standard 5 μm tungsten wire. The hot-wire anemometer in this study is small and can be attached to a probe support. In this study, uncertainty in the calibration curve of the hot-wire measurement is first checked. Here, Pitot tube velocimetry to obtain the calibration curves is carried out using two bell-type micro-differential pressure gauges. Then, the characteristics of the obtained velocity fluctuations and turbulence will be examined. Based on this relative velocity fluctuation RMS value, the effects of the obtained streamwise velocity fluctuation on output voltage are obtained by asymptotically expanding a calibration curve. Using these results, the intensity of the free-stream turbulence, which has a negligible effect on output voltage giving a calibration curve, is clarified.


Introduction
Free-stream turbulence without continuous energy input decays with time.In fundamental research, grid turbulence (e.g., [1]) is often used to experimentally reproduce such decaying turbulence.Recently, in addition to this conventional grid turbulence, multi-scale generated grid turbulence has also been considered [2][3][4][5].A hot-wire anemometer [6] is often used to experimentally investigate these fundamental turbulence phenomena.
Although the number of velocity components that can be measured by this hot-wire anemometer is small, it can accurately measure velocity fluctuations from low to high frequencies that are characteristic of turbulent flow phenomena.In recent years, particle image anemometry and laser anemometry have been widely used as experimental techniques, while high-accuracy numerical analysis and data-driven fluid dynamics can also be realized, but hot-wire anemometry is still used in the current research.For example, velocity fields in multi-scale generated grid turbulence have often been measured with hotwire anemometers in previous studies [4].The properties of spectra and structure functions of interest for this target can be measured with sufficient accuracy using a hot-wire anemometer.Improving the accuracy of the anemometer has also been investigated in recent years.Previous studies (e.g., [7]) have attempted to reduce the measurement uncertainty by using a compact hot-wire anemometer installed directly on the probe support.
When using a hot-wire anemometer, the output voltage needs to be calibrated using known velocity values.A calibration curve that relates velocity values to output voltage values is known as King's formula [6].Other calibration functions have been provided that extend King's formula.For example, King's formula with the exponent given as a function of flow velocity and the polynomial approximation to calculate flow velocity have been used in previous studies (e.g., [8]).The accuracy and uncertainty of the calibration curves obtained can directly affect the accuracy and uncertainty of the hot wire measurements.For example, a temporal change in the temperature of the air in the experimental environment can directly affect the output of a hot-wire anemometer [9].While this hot-wire calibration should preferably be performed in an ideal uniform flow, this study recognizes that there are often experimental situations where such conditions cannot be met.
The purpose of this study is to investigate measurement locations in the flow for which a constanttemperature hot-wire anemometer is calibrated.The focus of this study is on the velocity fluctuation field of the multi-scale generated grid turbulence, which has been the measurement target of the previous hot-wire anemometer experiments.In this study, the hot-wire anemometer calibration location is set up in the multi-scale generated decaying turbulence that is reproduced in the test section of the present wind tunnel experiment.This study attempts to evaluate the conditions that should be satisfied when calibrating hot-wire anemometers in a test section through which a turbulent main flow passes.Then, the velocity fluctuation intensity in the test section is measured.Based on these results, an asymptotic approximation of a calibration curve is used to address this research problem.

Experimental Methods
The blowout section in the present wind tunnel is square with a side of 0.4 m [10].The test section is connected to this blowout section.The cross section of the test section is a square with a side D of 0.4 m, just like the blowout section.The length of the test section in the direction of flow is 5 m.For this length of the test section, the effects of acceleration due to wind tunnel blockage have been validated to be negligible [11,12].This turbulence-generating grid is based on the fractal grid used in the previous studies [2][3][4][5].Here, for this multi-scale turbulence grid, the mesh with the smallest grid width is a conventional square grid.The number of steps in this turbulence grid is four.The basic element of the multi-scale grid is the square type in these previous studies.In this study, the characteristic length of this turbulence-generating grid is set to one side D of the test section.
The hot-wire anemometer system used in this experiment is shown in Fig. 1 (a).The hot-wire sensor is a 5 µm tungsten wire based on an I-shaped probe.This hot-wire anemometer is sufficiently compact to be installed on this support.The output voltage from this hot-wire anemometer is measured by a data logger (Keyence, NR-600) through a 10 kHz passive analog low-pass filter.The number of bits of A/D  conversion is 16 bits.The sampling frequency is 20 kHz and the sampling time is 60 seconds.Here, the value of the cutoff frequency [17] is higher than that of the sampling frequency.An analog oscilloscope (IWATSU SS7821) was also used in this study.The calibration positions in this wind tunnel experiment are set to x/D = 5.5 -10.
Here, a positioning green laser was used to verify the calibration positions as shown in Fig. 1 (b).The Pitot tube used in this study is a JIS type standard Pitot tube with a diameter of 4 mm.This Pitot tube is the same as the one used in the previous study [10].In this study, the Pitot coefficient of this Pitot tube was calibrated as a function of flow velocity.The pressure from the Pitot tube is measured by means of a submerged bell-type differential pressure gauge.The differential pressure gauge is calibrated with weights prior to each experiment.The calibration flow velocity values are set at approximately 3 m/s to 15 m/s.The free-stream velocity is set to be about 12 m/s.

Output voltage characteristics and calibration
This study first shows voltage time series data related to a velocity fluctuation displayed on the analog oscilloscope.Figure 2 (a) shows the instantaneous voltage time series displayed by the analog oscilloscope measured at the streamwise position where x = 4 m.In the figure, the oscilloscope display is obtained at approximately every frequency about 340 Hz.As shown in the figure, this study can see that an instantaneous voltage time series is displayed that closely resembles a typical turbulent velocity fluctuation of grid-generated turbulence.In addition, this voltage time series can be seen without unphysical voltage fluctuations that are visually associated with noise signals.
Figure 2(b) shows a power spectrum of the voltage fluctuation visualized by the oscilloscope.The value of the power spectrum decreases as the frequency increases shown in the figure.This result is similar to the typical spectral distribution of a velocity fluctuation in decaying homogeneous turbulence.Also shown in the figure is the power spectrum of the output voltage obtained when the flow around the hot wire is static with no free stream.This study assumes that this power spectrum corresponds to dark noise in an experimental measurement.As shown in the figure, the power spectrum of this dark noise is constant with the frequency.This result indicates that the output voltage of the dark noise corresponds to the white noise.As shown in the figure, the power spectrum does not decrease with increasing frequency in the high-frequency range.This result indicates that the power spectrum at high frequencies is not reduced by the dark noise.Calibration curves for converting an output voltage time series to a velocity time series are shown in Fig. 3.As shown in Fig. 3, the velocity values obtained increase with increasing values of the output voltage.In this experimental measurement, the calibration functions are measured both pre-and postmeasurements of the output at the downstream measurement point, where these results are shown as pre-and post-calibrations, respectively.As shown in the figure, the calibration functions are generally in good agreement between the pre-and post-calibrations.A seventh order polynomial function approximating these measured plots is also shown in the figure.This higher order polynomial function can represent the relationships between output voltage and velocity with sufficient accuracy.
The calibration curves shown in Fig. 3 are compared between the pre-and post-conditions in Fig. 4 (a).As seen in King's formula with generalized exponent value, the velocity is plotted on the horizontal axis for the calibration curves shown in the figure because the calibration function with velocity as a variable has a clearer physical meaning.As shown in the figure, there is a slight difference between the calibration curves for the pre-and post-conditions.The following calibration function based on King's law is fitted to the measurement plots in this study to investigate a factor causing this difference. ), where DA = (Apost -Apre)/Apre, DB = (Bpost -Bpre)/Bpre, and Dn = (npostnpre)/npre. (2) Here the subscripts, pre and post, denote the quantities for the pre-and post-calibrations of the present measurements, respectively.As shown in the figure, the above calibration function is able to adequately reproduce the obtained curves.This calibration function is specified by the values of the three included coefficients.The values of the coefficients are compared between the pre-and post-conditions in Fig. 4 (b).As shown in the figure, the relative deviations in the values of the coefficients B and n between the pre-and post-conditions are sufficiently small.On the other hand, the deviation of the coefficient A between these conditions is significantly larger than that of the other two coefficients.Here, A is often Figure 3. Calibration curves for calculating velocity from output voltage.These curves are obtained for pre-and post-measurement conditions.The measured plots are approximated by a seventh-order polynomial.referred to as the calibration drift [13][14][15][16].These results imply that the calibration curve has changed between the pre-and post-conditions due to the change in calibration drift.
This study focuses on the asymptotic expansion of a calibration curve to consider the results shown in the figure.The instantaneous voltage and instantaneous velocity are given as follows, using the characteristic velocity and the voltage value as the characteristic value given by this velocity.For the instantaneous output voltage E and instantaneous velocity  ", Reynolds decomposition is applied in this study.
E( ") = áE( ")ñ + e. ( Here, á ñ denotes ensemble average and voltage fluctuation áeñ = 0.When the velocity fluctuation is the voltage fluctuation can be related to the velocity fluctuation by using the above equation with Taylor expansion around a velocity taken to be Uo as follows. Here áE( ")ñ = E(Uo) due to the small intensity.As shown in this relationship, the velocity fluctuation is not related to the voltage fluctuation by the calibration curve itself, so the first-order derivative of the calibration curve is used to relate the voltage fluctuation to the velocity fluctuation.the PDF obtained in this study can be seen to be well approximated by a Gaussian distribution.This result is in qualitative agreement with those shown in previous studies.

Velocity fluctuation characteristics
As shown in the instantaneous relative velocity time series results, the amplitude of the instantaneous relative velocity is in the range of 5-6% relative to the free stream.Specifically, the RMS  value of the instantaneous relative velocity fluctuation at this calibrated position can be obtained as 0.021.Based on this result, the effect of free-stream turbulence on the Pitot pressure at the calibration position is verified in this study.The effect of the freestream turbulence on the observed value of the Pitot pressure is in the order of the intensity of the fluctuation, not the RMS value of the velocity fluctuation of this turbulence.The effect seen in this study is estimated to be about 0.04% and can be considered sufficiently small.Therefore, the RMS values obtained in this study indicate that the effect of this free-stream turbulence on the Pitot pressure values observed at the calibration location in this study is sufficiently small.
Figure 6 shows a variation in the streamwise direction of an autocorrelation function at downstream positions, x/D = 8 to 10.Here, the horizontal axis is normalized to the width D of the test section, which is the characteristic length of this study.As shown in the figure, the autocorrelation function obtained in this study takes the value of unity at the origin and also decreases as the distance between two points increases.For a large distance between two points, the value of the autocorrelation function asymptotically approaches zero as the distance increases.These results are consistent with those widely observed for the autocorrelation function in grid turbulence.These results validate that a typical grid turbulence field is reproduced in the test section of this study and is satisfactorily measured by the present hot-wire anemometer.In the figure, the present study can see that the distribution of the autocorrelation function gradually expands as the normalized distance between two points increases with the position around 0.3 in the streamwise direction.
By integrating the autocorrelation function, the integral scale can be given as follows.A variation of the integral scale in the flow direction is shown in Fig. 6 (b).Here, the integral scale in this experiment is normalized by the width of the test section of the wind tunnel, similar to the distance between two points.As shown in the figure, the value of the integral scale increases in the streamwise direction.This study can see that the value of this normalized integral scale downstream is at most 0.1.This figure also shows the distribution of the RMS of the velocity fluctuation non-dimensionalized by the characteristic velocity in the streamwise direction.The decreasing velocity fluctuation RMS distribution with increasing integral scale distribution for the streamwise direction shown in the figure is consistent with characteristics widely found in homogeneous decaying turbulence, as implied by the form of the invariants of decaying turbulence [18].

Examination using a calibration function
The effect of streamwise turbulence at the calibration position on the calibration results is examined.The asymptotic expansion of the calibration curve presented in the previous paragraph is used here [9].The Taylor expansion of the output voltage around the calibration velocity value gives Here, different from the previous discussion, the above equation includes a second order term in addition to the first order term.Taking the average of the above equation, the following is obtained.
As shown in the above equation, if the velocity fluctuation intensity is not negligible, the mean value of the output voltage may not match the voltage value at the calibrated velocity.The following can be obtained by modifying the above equation.
Here, a value of calibration drift A is generally assumed to be sufficiently smaller than the voltage at the calibration flow velocity, specifically, E(Uo) >> A. As shown in the above equation, the second term on the right side of the above equation is clearly characterized by the relative velocity fluctuation intensity.In this study, the magnitude of the relative velocity fluctuation intensity is 0.04%, so the second term on the right-hand side is negligibly smaller than the first term on the right-hand side.This result indicates that the effect of freestream turbulence on the calibration results in this study is negligible.
The results for the output voltage fluctuations of the asymptotically extended calibration curve are discussed based on the observation that the PDF of the velocity fluctuation can be approximated by a  Gaussian.The first and second order terms of the calibration curve, denoted as e' here, are extracted as follows.
By squaring both sides of the above equation and taking the ensemble mean, the following is obtained.
Here, the following relationship is used for the velocity variation to follow the Gaussian: áu 3 ñ = 0 and áu 4 ñ = 3 áu 2 ñ 2 .By dividing both sides of this equation by the voltage value of the calibrated velocity, the following is obtained using Eqs.( 3) and ( 4). where The coefficients of the first and second terms on the right-hand side are given by the differential coefficients of the calibration curve.

Discussion and Conclusion 4.1 Discussion
As seen in this study, the effect of freestream turbulence on the calibration output voltage is on the order of the intensity of the relative velocity fluctuations, and therefore the intensity of the relative velocity fluctuations should be sufficiently small at the calibration site.In multi-scale generated turbulence, there may be upstream velocity fluctuations that result in RMS values of 5-10% relative to the mean velocity.In this case, the relative velocity fluctuation intensity may reach 0.25 to 1%.This estimate indicates that the hot-wire anemometer should not be calibrated in the upstream region of multi-scale generated turbulence.The study considers that cases where hot-wire anemometers are calibrated in freestream flows with turbulence can also be found in flows with boundary layers.In cases where the effects of freestream turbulence on boundary layer characteristics are being examined, or where the freestream varies with time, the results of this study can be used to estimate the effects of changes or variations in the flow on the calibration performance.
The previous studies [13][14][15][16] have shown recalibration as a calibration method for hot-wire anemometers.This method requires recalibration using the freestream velocity at specified time intervals.In addition, previous studies have recalibrated using the freestream of a boundary layer to measure the velocity statistics of the layer.This study considers that the results of this study can provide information for determining locations where the recalibration operation should be performed.The results of this study can be used to characterize locations in the flow field to be measured where the effect of freestream turbulence on the calibration results is sufficiently small.In the recalibration method, the accuracy of a calibration curve during the recalibration procedure directly affects the measurement results of the velocity statistics.In this study, the results obtained can be used to contribute to validating the accuracy of the recalibrating method.

Conclusion
The effect of freestream turbulence on calibration results when calibrating the hot-wire anemometer in the test section was addressed in this study.The calibration position of the anemometer was set in the downstream region where the multi-scale generated grid turbulence reproduced in the test section decays.The hot-wire anemometer in this experiment is small to be mounted in the winged support.The hot-wire probe was the I-shaped probe based on a 5 µm tungsten wire.The mean velocity, also measured at this calibration position, was obtained using the Pitot tube with the calibrated Pitot coefficient.
In this study, the output voltage was first validated by the flow visualization result using the analog oscilloscope and the spectra.The calibration curves were measured pre-and post-voltage measurements at the calibration position.The deviation between these two calibration curves was sufficiently small.This small deviation was due to the calibration drift.The time series of the velocity variation at this calibration position was examined and could be approximated by the Gaussian.From this time series, the relative velocity fluctuation RMS at the calibration position was found to be about 0.02.
Since hot-wire anemometers are still often used in addition to numerical analysis and particle image anemometry to examine turbulence fields in multi-scale generated turbulence, the results of this study can contribute to the study of multi-scale generated turbulence fields.There may be few previous studies of multi-scale generated turbulence that have investigated characteristics in the upstream compared to those in the downstream.Velocity fields of multi-scale generated turbulence in the upstream region cannot be approximated by Gaussian and may have characteristics of high intermittency.Therefore, this study considers that the characteristics of multi-scale generated turbulence in the upstream region should be examined based on the results of this study.

Figure 1 .
Figure 1.Schematic figures related to the hot-wire anemometer in the present wind-tunnel apparatus.(a) overview; (b) diagram of the hot wire probe shown with the Pitot tube for calibration.

Figure 2 .
Figure 2. Measured voltage signal at downstream x/D = 10.(a) An example of visualization of a voltage fluctuation using the present oscilloscope.(b) Power spectrum of the voltage fluctuation Se at this position.

Figure 5 (
a) shows a time series of instantaneous relative velocities measured at the calibration position x/D = 10.The time series is shown over the first 0.1 seconds of the sampling time.Furthermore, the time series is observed to consist of waves from low to high frequencies.The probability density function (PDF) of the instantaneous relative velocity time series is shown in Fig. 5 (b).As shown in the figure,

Figure 4 .
Figure 4. Results for the deviation between the calibration curves under the pre-and postmeasurement conditions.(a) Results of fitting the calibration equation based on King's formula to the measured plots.

Figure 5 .
Figure 5.Time series of the normalized velocity at downstream x/D = 10 (a) and a probability density function (PDF) for its fluctuating component (b).

Figure 6 .
Figure 6.Results of validation of velocity fluctuation fields in downstream areas using autocorrelation functions and integral scales.(a) Autocorrelation function in the downstream region.Here, the distance between two points is normalized by the representative length D. (b) Streamwise profiles of integral scale and values of the relative velocity fluctuation RMS in the downstream region.