# Simplified Calibration Method for Constant-Temperature Hot-Wire Anemometry
## 3 Results  
Figure 4a presents an example result for Run 1 of measured boundary layer profiles made by raw hot-wire anemometry data (HW(Raw Signal)), temperature-corrected hot-wire anemometry data (HW(Corrected)), and reference pitot-tube-measured data (Pitot). Calibrated data based on Equation (1) is also plotted for comparison denoted as HW(By Equation(1)). The nondimensional velocity normalized by the freestream velocity (in horizontal axis) is plotted against the height from the wall surface (in vertical axis). Each hot-wire data point was averaged over 5 s and the error bar shows 1\(\sigma\) over the averaging duration. Run 1 had the largest temperature variation (i.e., 15.1 \({}^{\circ}\)C) in this study. A large temperature variation during the wind tunnel operation is seen from the raw hot-wire anemometry data (HW(Raw Signal)). The temperature variance in freestream temperature, which was seen in Figure 3, became longer for longer wind tunnel operation duration. The profile calibrated by Equation (1): HW(By Equation (1)) shows large discrepancy between it and that by reference (Pitot). In this case, which had the large temperature variation, a proposed method was used instead of Equation (1). By searching for a \(T_{w}\) value that meets Equation (3), the hot-wire data will be corrected with regard to the temperature variance. The increment for \(T_{w}\) search was 1 \({}^{\circ}\)C as mentioned in Section 2.3. From Figure 4a, the corrected distribution (HW(Corrected)) agreed well with that obtained from the pitot-tube measurement (Pitot). The \(T_{w}\) for this case was found to be 111 \({}^{\circ}\)C with a maximum error of 1.8%.  
Figure 4b presents an example result for Run 2 shown as Figure 4a. The temperature variation in this case was negligibly small (less than 1 \({}^{\circ}\)C) as seen in Figure 3. Differently from Run 1 (Figure 4a), almost no temperature variation or temperature dependence in the raw signal profile (HW (Raw Signal)) is seen. The profile calibrated by Equation (1) denoted as HW (By Equation (1)) shows better agreement  
Figure 4: Representative comparison for velocity profiles obtained by hot-wire anemometry (raw, calibrated by King’s law, and temperature-corrected by current method) and pitot-tube measurement.  
This method can be applied to other cases as well. Figure 7 is a histogram showing uncertainties calculated as a velocity difference between the pitot-tube measurement data and the temperature-corrected hot-wire anemometry data measured at the same distance from the wall. The wire temperature for each case was the optimal temperature found by the best-fit approach proposed in this study. This uncertainty was defined by Equation (4). A total of 200 data points were obtained from eight wind tunnel runs with 25 probing at each case and were plotted. The mean and 1\(\sigma\) values for those error values were plotted as well. As seen in Figure 7, the mean error level was 0.48 and with 95% certainty for the error level stayed below 1.34%.  
\[\delta_{H_{i}}=\frac{\left|U_{pitot_{i}}-U_{HW_{i}}\right|}{U_{pitot_{i}}} \tag{4}\]  
Those results indicate that the current temperature correction method that is a slight modified form of Equation (2) could be applied when reference velocities with temperatures are known regardless of the data points used for the hot-wire anemometry. Though the uncertainty level of the velocity range was found to be between 20 and 80 m/s, the proposed method on the temperature correction was validated. These results also indicate that the proposed approach could be applied regardless of velocity values and individual differences in hot-wire probes.  
Figure 6: Accuracy of temperature correction by various wire temperatures.  
Figure 7: Uncertainties inherent in the proposed calibration method plotted against the error of velocity calibration.