General setup
Like all IR-based techniques, the approach described here requires a coating of the surface with low thermal conductivity and high emissivity. By heating the respective surface with a short light pulse (generated, e.g., by a flash lamp or a laser), the topmost layer of the coating is heated up a few degrees. Two-dimensional areas can be illuminated at once so that no scanning of the surface is necessary. After the light source has been switched off, a high-speed infrared camera is used to detect the temperature decline for every pixel. This temperature decline is evaluated in a post-processing algorithm resulting in a quantity proportional to the heat transfer coefficient and the wall shear stress. Figure 1 schematically explains the measurement setup.
In IR-based measurements, different parameters can interfere with the measured signal and lead to inaccurate results. These include irregular thermal coatings due to varying thicknesses, heat loss into the blade due to non-ideal insulation, inhomogeneous heating of the used light source and interfering reflections due to surrounding components. For the proposed measurement technique, these effects can be distinctively reduced by a novel post-processing method and a reference measurement which is performed without any airflow and subtracted from the measurement with flow. Both measures significantly improve data quality and will be theoretically motivated in the following.
Theory and measurement data analysis
To analyze the data and to determine a quantity proportional to the heat transfer coefficient and the wall shear stress, the simplified model of an ideal thermal insulator is used. The temperature rise (\(T^{\prime} = T - T_{\infty }\)) directly after the heating pulse is given as
$$T^{\prime}_{0} = \frac{{q_{\text{pulse}} }}{{c_{\text{e}} \rho_{\text{e}} h}},$$
(1)
where \(q_{\text{pulse}}\), \(c_{\text{e}}\) and \(\rho_{\text{e}}\) are the energy density of the heating pulse, the specific heat and the density of the coating, while \(h\) denotes the penetration depth of the light pulse. Due to Newton’s law of cooling, the heat flux from the heated surface to the air flow is given by
$$\dot{q}_{\text{w}} = - \alpha T^{\prime},$$
(2)
and therefore, the differential equation for the simplified model of an ideal insulator can be written as
$$\dot{T}^{\prime} = - \frac{\alpha }{{c_{\text{e}} \rho_{\text{e}} h}}T^{\prime},$$
(3)
with the solution
$$T^{\prime}(t) = T^{\prime}_{0} \exp \left( { - \frac{\alpha }{{c_{\text{e}} \rho_{\text{e}} h}}t} \right),$$
(4)
where \(\alpha\) represents the heat transfer coefficient.
In the case of an ideal insulator, the heat transfer into the flow can be expressed as
$$\alpha = - c_{e} \rho_{e} h\varLambda$$
(5)
with \(\varLambda\) as the temperature decline rate. After thermal excitation, \(\varLambda\) is constant and can be determined by the following expression
$$\varLambda = \frac{1}{\Delta t}\ln \left( {\frac{{T^{\prime}_{m + 1} }}{{T^{\prime}_{m} }}} \right) \approx \frac{{2\left( {T^{\prime}_{m + 1} - T^{\prime}_{m} } \right)}}{{\Delta t\left( {T^{\prime}_{m + 1} + T^{\prime}_{m} } \right)}},$$
(6)
where \(\frac{1}{\Delta t}\) is the frame rate of the camera and \(m\) is the picture index, while the logarithm was approximated by its series expansion. For subsonic air flows with Prandtl number \(Pr \approx 1\), the Reynolds analogy factor
$$s = \frac{{c_{\text{f}} }}{2St}$$
(7)
is approximately equal to unity and constant (Incropera and De Witt 1985; Luca et al. 1990). \(c_{f}\) denotes the coefficient of friction while \(St\) is the Stanton number. The coefficient of friction in the flow is given as
$$c_{\text{f}} = \frac{{\tau_{\text{w}} }}{{\frac{\rho }{2}U_{\infty }^{2} }},$$
(8)
with \(\tau_{\text{w}}\) as wall shear stress, \(\rho\) as the density of the fluid, and \(U_{\infty }\) as the unperturbed velocity of the fluid (Schlichting and Gersten 2006). The heat transfer \(\alpha\) and the Stanton number are related by
$$St = \frac{\alpha }{{\rho c_{\text{p}} U_{\infty } }},$$
(9)
with \(c_{\text{p}}\) as the isobar heat capacity of the fluid. Equations (5), (7), (8) and (9) can be combined to the following expression
$$\tau_{\text{w}} = \frac{\alpha }{{c_{\text{p}} }}sU_{\infty } = \varLambda \frac{{c_{\text{e}} \rho_{\text{e}} h}}{{c_{\text{p}} }}sU_{\infty } ,$$
(10)
which shows that the wall shear stress, the heat transfer coefficient and the temperature decline rate are proportional if the coating resembles an ideal insulator. For data analysis, Eq. (6) is used to calculate \(\varLambda\) for every pixel of subsequent thermograms and directly relate it to \(\alpha\) and \(\tau_{\text{w}}\). By definition, \(\varLambda\) is independent from the absolute temperature level. Consequently, inhomogeneities of the initial heat pulse are of minor importance and reflections of surrounding components can be strongly reduced by the usage of the proposed analysis method.
Numerical simulations
With thermal conductivities of about 0.1 W/(m K), the condition of an ideal insulator is not achievable even with modern coatings. Especially on complex shapes which often lead to a varying thickness of the coating, the heat loss into the blade cannot be neglected. To analyze the sensitivity of the present approach to this effect, a simplified 1D numerical simulation was used, which calculates the heat transport processes for given heat transfer coefficients and heat conduction rates. Furthermore, thermal coatings consisting of more than one layer (e.g., high emissivity coating and an additional insulating layer) with differing material parameters and thicknesses can be simulated to determine the temperature decline on the surface.
An implicit simulation based on Bender–Schmidt’s method is used which approximates the heat conduction equation at every grid point (Golub and Ortega 1995; Jaluria and Torrance 2003). Only the heat transport perpendicular to the surface is taken into consideration. This approximation is justifiable under several circumstances, namely an approximately constant temperature distribution on the surface, very small thicknesses of the thermal coating layers or if the spatial resolution of the measured image (defined by the optics of the infrared camera and the detector) is by far coarser than the distance given by the propagation velocity of the heat waves through the thermal coating within the observed time frame.
This numerical method was used to calculate the temperature decline rate versus time for a three-layer thermal insulation system (Fig. 2, left). At 0.07 s, where \(\varLambda\) is approximately constant, several \(\varLambda\) values were evaluated for different heat transfer coefficients between 0 and 2800 W/(m2 K). By plotting the respective \(\varLambda\) values versus heat transfer (Fig. 2, right), it can be shown that the relation between heat transfer and \(\varLambda\) is approximately linear for small variations in \(\varLambda\) and \(\alpha\), which corresponds to the described theory of the ideal insulator [see Eq. (5)]. The major deviation to this theory lies in the offset of the linear relation since in reality heat loss due to conduction into the material also occurs at \(\alpha = 0\). To take this into account, a measurement in a reference situation without flow (\(\alpha = 0\)) must be subtracted from a measurement with flow: \(\varLambda - \varLambda_{\text{ref}}\). This reduces interfering effects such as variation in thickness of thermal coatings and inhomogeneous illumination while preserving the linear relation between \(\varLambda\) and \(\alpha\).