#### 2.2. Principle and Performance

The flexible hot film shear stress sensor works on the basis of the temperature-resistance characteristic of nickel thermistor, that is, the nickel thermistor’s resistance changes when the temperature changes. The temperature coefficient of resistance (TCR) is defined as the increase of resistance

$R$ by percentage for one degree increase of temperature

$T$.

$R$ changes by

$\mathrm{d}R$ when

$T$ changes by

$\mathrm{d}T$,

${R}_{0}$ is the cold resistance of the flexible hot film shear stress sensor at room temperature

${T}_{0}$. TCR can be written as follows:

where

α is TCR, reported in ppm/°C. From Equation (1),

$R$ can be calculated when the temperature changes.

When a fluid with a certain velocity flows through the flexible hot film shear stress sensor, it will take heat away from the thermistor, and then the temperature of the thermistor changes. As a result of the thermistor’s temperature-resistance characteristic, the resistance of the thermistor will change accordingly. The heat loss is related to the shear stress. Based on the energy balance theory, the heating energy arising from the thermistor equals the energy loss to the ambient flow and substrate. The convective transfer of heat from the thermistor to the ambient flow is a function of the flow velocity. The heating power

$P$ and the shear stress

${\tau}_{w}$, follow the relation:

where

${T}_{t}$ and

${T}_{f}$ are the temperature of the thermistor and the flow, both reported in K;

$P$ is reported in W;

${\tau}_{w}$ is reported in Pa;

$A$ and

$B$ are calibration constants.

When the flexible hot film shear stress sensor works in the circuit, the heating power

$P$ can be written as:

where

$E$ is the voltage across the sensor, in V. So, Equation (2) becomes:

After calibration, from Equations (1) and (4), the wall shear stress ${\tau}_{w}$ can be inferred.

The flexible hot film shear stress sensor is operated in constant temperature (CT) mode when calibrating the constants. For the CT mode, the resistance

$\mathrm{R}$ is unchanged,

${T}_{t}-{T}_{f}$ is constant. Equation (4) then becomes:

where

${A}_{\mathrm{CT}}$ and

${B}_{\mathrm{CT}}$ are calibration constants in CT operation.

$E$ can be read from the oscilloscope connected to the flexible hot film shear stress sensor. In the relationship curve between

${E}^{2}$ and

${\tau}_{w}^{1/3}$, the intercept is

${A}_{\mathrm{CT}}$ and the slope of the curve is

${B}_{\mathrm{CT}}$.

${B}_{\mathrm{CT}}$ is called the sensitivity of the flexible hot film shear stress sensor. According to Equation (5),

${B}_{\mathrm{CT}}$ can be found in terms of the output voltage and the calculated shear stress [

1,

13].

The relationship between shear stress and the stream-wise pressure distribution can be expressed as:

where

${P}_{x}$ is the local pressure, in newton;

$h$ is the half height of the wind tunnel and

$x$ is the stream-wise coordinate, both reported in m [

15]. Pressure taps are used to obtain the stream-wise pressure gradient.

${\tau}_{w}$ can be calculated by Equation (6).

Time constant is used to characterize the sensor response to a step input. It is obtained by feeding square wave into the flexible hot film shear stress sensor of CT circuits. When a square wave passes through, the transient response of voltage on the thermistor is captured, from which the time constant is deduced. The time constant is determined from the experimental curve shown in

Figure 3 [

13]. The approximate relation between cut-off frequency

${f}_{c}$ and time constant

${t}_{c}$ is: