Turbulent heat flux measurement in a non-reacting round jet, using BAM:Eu²⁺ phosphor thermography and particle image velocimetry

Abstract

Turbulent mixing is highly important in flows that involve heat and mass transfer. Information on turbulent heat flux is needed to validate the mixing models implemented in numerical simulations. The calculation of turbulent heat fluxes requires instantaneous information on temperature and velocity. Even using minimally intrusive laser optical methods, simultaneous measurement of temperature and velocity is still a challenge. In this study, thermographic phosphor particles are used for simultaneous thermometry and velocimetry: conventional particle image velocimetry is combined with temperature-dependent spectral shifts of BAM:Eu²⁺ phosphor particles upon UV excitation. The novelty of this approach is the analysis of systematic errors and verification using the well-known properties of a heated turbulent jet issuing into a low velocity, cold coflow. The analysis showed that systematic errors caused by laser fluence, multiple scattering, or preferential signal absorption can be reduced such that reliable measurement of scalar fluxes becomes feasible, which is a prerequisite for applying the method to more complex heat transfer problems.

Appendices

Appendix 1

For an optimal choice of central wavelengths and filter bandwidths in the ‘blue’ and ‘red’ channels, a trade-off between temperature sensitivity and signal strengths determining temperature precision uncertainty has to be considered. At the expense of signal intensity for narrower bandwidth and specific central wavelength positions the sensitivity can be enhanced and vice versa.

Eq. (1) introduced the signal ratio $= \frac{{S_{ph}^{blue} }}{{S_{ph}^{red} }}$. Assuming shot and readout (N _cam ) noise only, as well as statistical independency of noise in both channels, the precision uncertainty of the intensity ratio can be approximated by

$$\sigma_{r} = r\sqrt {\frac{1}{{N_{ker} }}\left[ {\left( {\frac{{\sqrt {KS_{ph}^{blue} + N_{cam}^{2} } }}{{S_{ph}^{blue} }}} \right)^{2} + \left( {\frac{{\sqrt {KS_{ph}^{red} + N_{cam}^{2} } }}{{S_{ph}^{red} }}} \right)^{2} } \right]}$$

(5)

K is the conversion factor of the analog–digital converter, and N _ker is the number of pixels in the averaging kernel. To calculate σ _r , camera readout noise and signals in ‘blue’ and ‘red’ channels are needed in units of ‘counts.’ For the cameras, the measured readout noise N _cam was 3.5 counts. To determine the signal level, S _ph , phosphorescence images were recorded without interference filters. Using the relative emission intensities shown in Fig. 1, absolute and wavelength-dependent signal levels can be derived for various temperatures. These absolute and wavelength-dependent signal levels are valid only for this specific setup and vary when changing f#, transmission efficiencies of the camera lenses and quantum efficiencies of the cameras. By assuming ‘virtual’ transmission characteristics of ‘blue’ and ‘red’ interference filters and integrating them across the ‘virtual’ spectral bands, signal levels in the channels can be estimated.

Based on these assumed ‘virtual’ transmission characteristics of both filters, a ‘virtual’ calibration curve that is parameterized by a second-order polynomial (T = ar ² + br + c) can be determined. The sensitivity of the correlation between ‘temperature T’ and ‘ratio r,’ $\frac{dT}{dr}$, is the first derivative of the calibration curve (2ar + b). The experimental precision, σ _r , can then be transformed to a temperature uncertainty by $\sigma_{T} = \frac{dT}{dr}\sigma_{r}$.

This equation shows that the temperature uncertainty depends both on the sensitivity of the calibration curve, $\frac{dT}{dr}$, and the precision in measuring the intensity ratio, σ _r . Using this equation, different choices of central wavelengths and bandwidths of both filters were evaluated. Limited by the availability of commercial interference filters a best compromise of sensitivity and signal strengths has been selected. The transmission curves of the so selected interferences filters used for ‘blue’ and ‘red’ channels are included to Fig. 1.

Appendix 2

In the particle locating algorithm introduced in Sect. 3.3, a proper choice of threshold Mie intensity was required to identify areas with no particles present. Increasing the threshold removed camera noise and spurious reflections that appeared at very low signal levels. However, increasing the threshold at the same time reduced the number of particles identified in a specified interrogation volume (here 170 × 170 µm). Correspondingly for instantaneously recorded images, the phosphorescence signal integrated across the interrogation volume relative to the number of particles contained in the interrogation volume (yielding ‘signal per particle’) varied by varying the threshold intensity. To identify an appropriate intensity threshold, a parametric sensitivity study was conducted. Therefore, the standard deviation of the ‘signal per particle’ was evaluated for a homogenous temperature distribution in dependence of the Mie intensity threshold. The threshold was set to 500 counts; a value which minimizes the standard deviation of ‘signal per particle’ and the effect of deleting real particles.

In the next step, the sensitivity of the number particles per interrogation area on the measured temperature was evaluated. Interrogation volumes that contained at least (a) one, (b) two, or (c) three particles were considered. Figure 12 shows mean radial temperature profiles for the three particle number threshold values applied to the same set of data. Additionally, thermocouple measurements are compared to profiles from phosphor thermometry. Differences between profiles are shown in the insert.

Fig. 12

Mean radial temperature profiles at x/d = 1 obtained using 170 × 170 µm wide interrogation volumes that contained at least 1, 2, or 3 particles, denoted by symbols open circle: 1, plus sign: 2, open square: 3. Thermocouple measurements are denoted by triangles. Relative temperature differences are shown in comparison with a mean temperature profile where the threshold has been set to zero

Varying the minimum number of particles contained in each interrogation volume (before final binning to 510 × 510 µm) from 1 to 3 changes the temperature within the shear layer by less than 10 K. In all other regions, the effect was even weaker. This indicates that the seeding density in the coflow of this study was low enough that the multiply scattered light did not bias the temperature of regions containing only few particles. However, in cases, where the coflow seeding density is required to be higher than in this study, our filtering procedure will help to reduce the bias by multiple scattering for locations with few particles and correspondingly low phosphorescence intensity. There was a remaining difference of 10 K in the shear layer with increasing particle number threshold, where hot jet and cold coflow are being mixed. With the higher particle number threshold, it is more likely to represent the temperature from the jet rather than the coflow. Therefore, the temperature in the shear layer increases with increasing particle number threshold because it better represents the jet temperature. Deviations in thermocouple measurements in the shear layer are up to 30 K and are attributed to differences in location and spatial resolution of both measurement techniques.