# Simplified Calibration Method for Constant-Temperature Hot-Wire Anemometry
## 1 Introduction  
Hot-wire anemometry for flow velocimetry has been widely used in wind tunnel experiments in various flow applications [1]. Using its high-resolution measuring capability with both a temporally and spatially high frequency response, statistical turbulent features in the unsteady flow process can be extracted. In addition to past studies, recent progress of the measurement technique facilitates wider applications such as supersonic unsteady flow measurements for a mixing study [2], superfluid measurements [3], subsonic compressible turbulent flow measurements with analytical approach [4], and unsteady wake measurements behind a wind turbine [5]. Briefly describing the principle of measurement by hot-wire anemometry, it uses a small, electrically heated wire exposed to a flow for the purpose of measuring a flow property. The flow property to be measured is usually the flow velocity. The hot-wire probe consists of a thin wire with prongs to support the wire. While measurement, the wire is heated by a current that passes through it. The heat transfer coefficient between the thin heated wire and the surrounding fluid medium changes as the change of the properties of moving fluid. Thus, the flow property is obtained from the change of heat transfer coefficient of the wire. There are two basic operating modes on hot-wire anemometry: the constant temperature mode and the constant current mode. The former maintains the wire temperature (\(T_{w}\)) at a constant value throughout the operation by means of an electrically controlled feedback signal by a Wheatstone bridgecircuit and the latter maintains the current at a constant value. This paper focuses on the constant temperature operating mode. In the constant temperature operating mode, the resistance of the probe's sensor material changes with changing temperature of the environmental fluid medium. The output electric signal from the hot-wire sensor also changes according to both freestream velocity and freestream temperature since the wire element is sensitive to environmental temperature and flow compositions such as velocity. Therefore, it is necessary to convert the measured output voltage signal to a flow velocity by a proper temperature correction method for quantitative velocimetry. In case, the flow temperature variation throughout the experimental operation duration (e.g., wind tunnel running duration) is negligibly small, the output voltage signal can be simply considered as a function of only flow velocity. In this case, King's law [1; 6; 7], which is expressed as Equation (1), is employed to convert the voltage signal to the flow velocity. \[E^{2}=a+bU^{n};\;n=0.45 \tag{1}\]  
Here, constants \(a\) and \(b\) are determined by a number of known flow velocities with an ordinary least squares fit for \(a\) and \(b\). Differently, in case the temperature variation in the flow throughout the running operation duration is relatively large, the temperature dependence can no longer be ignored and the measured voltage signal becomes a function of both flow velocity and temperature. In this case, a temperature correction method suggested by [PERSON] et al. [8], which is expressed in Equation (2), is useful. \[U=\
u\cdot f\left(\frac{E^{2}}{k\Delta T}\right)=\
u\cdot f\left(\frac{E^{2}}{ k(T_{w}-T_{a})}\right) \tag{2}\]  
where \(f\) denotes a functional dependence, \(\
u\) is the kinematic viscosity, and \(k\) is the thermal conductivity of air. Since both variables are dependent on the freestream temperature \(T_{a}\), \(\
u\) can be found by using [PERSON]'s law and air density, and \(k\) can be obtained from an empirical model [9]. Note that this method was proposed for the subsonic flow regime [8]. This technique is useful when only a single velocity calibration with a known \(\Delta T\) is sufficient, but it needs to know the wire temperature \(T_{w}\). A slight modification of the method is needed when the wire temperature is unknown.