# Simplified Calibration Method for Constant-Temperature Hot-Wire Anemometry
## 2 Measuring Technique, Equipment, and Data Reduction
### Alternative Method of Temperature Correction  
Figure 3 shows the freestream temperature change due to wind tunnel operation as the probe position (\(z\)) changes. The horizontal axis is the measuring number, or nondimensional operation time. The higher the measuring number, the longer the wind tunnel operation is. Temperature variations seen in four representative wind tunnel runs are presented. Obviously, the freestream temperature increased gradually as the wind-tunnel operation time increased. This observation is commonly applied to all other runs. Additionally, included are cases with relatively small temperature variations (i.e., less than 1 \({}^{\circ}\)C throughout the experiment), such as Run 2 in Figure 3. The temperature variation for other cases ranged from 6 to 15 \({}^{\circ}\)C. Thus, a variety of temperature variation ranges were considered in this study. Since the resistance of the hot-wire probe is a function of both freestream velocity and temperature, an appropriate temperature correction must be made to those wide variety of temperature variation data for quantitative analysis.  
As described in the Introduction, it is necessary to know the wire temperature \(T_{w}\) to find the velocity by solving Equation (2), and this wire temperature is key to temperature correction for obtaining the quantitative velocity. When the true wire temperature is unknown, an alternative temperature correction with a slight modification of the original temperature correction method [8] can be used using the best-fit approach to quantitatively calibrate the velocity without knowing the wire temperature.  
The functional dependence of velocity on the voltage signal (found in Equation (2)) was established by measuring velocities by the pitot-tube measurement. Freestream temperatures were also measured upon changing the probe height in the \(z\)-direction. The wire temperature (\(T_{w}\)) in Equation (2), which was unknown, was parametrically changed from 70 to 200 \({}^{\circ}\)C with an increment of 1 \({}^{\circ}\)C, the range of those temperatures were considered to find a possible wire temperature. Then, its value that makes the difference of the \(U\) velocity profiles obtained by the hot-wire anemometry data (Equation (2)) and the pitot-tube measurement data was sought to become zero or minimum. This iteration method is a similar but simpler method than an output error method, which has been widely applied and validated in flight control study [11; 12; 13], for obtaining the best fit (Equation (3)). An increment of 1 \({}^{\circ}\)C for \(T_{w}\) search was used in this study. It should be noted that the current method depends on the initial guess (e.g., 70 \({}^{\circ}\)C in this study) as well as the temperature increment. If the initial guess for the wire  
Figure 3: Freestream temperature variation plotted against measuring number at different heights from the wall during wind tunnel operation for four different runs.  
temperature is sufficiently low with small temperature increment, the optimal wire temperate can be found with better accuracy by Equation (3) but computation load may increase.  
\[\delta_{total}=\sum_{i=1}^{N}\delta_{i}=\sum_{i=1}^{N}\left|U_{pitot_{i}}-U_{HW_{i }}\right|=f(T_{w}) \tag{3}\]  
Thus, this method searches \(T_{w}\) values to find the minimum \(\delta_{total}\) to facilitate the temperature correction of the voltage signal obtained from the hot-wire anemometry without knowing the actual \(T_{w}\) value.