Measurement of instantaneous fully 3D scalar dissipation rate in a turbulent swirling flow

Abstract

This paper describes the measurement methodology for quantifying the instantaneous full 3D scalar dissipation rate (SDR or $\chi$) in order to characterize the rate of mixing. Measurements are performed in a near field of a jet-in-swirling-coflow configuration. All three components of $\chi$ are measured using a dual-plane acetone planar laser-induced fluorescence technique. To minimize noise, a Wiener filtering approach is used. The out-of-plane SDR component ($\chi _3$) is validated by assuming isotropy between axial and azimuthal components of SDR. An optimum laser-sheet separation distance ($\varDelta s$) is identified by comparing the SDR components on the basis of instantaneous, mean, and probability density function data. The in-plane resolution needs to match the Batchelor scale ($\lambda _B$) for the central difference scheme-based SDR deduction. However, the out-of-plane resolution, $\varDelta s$, requirement is different owing to the use of two-point difference based SDR and systematic biases. The optimum $\varDelta s$ is found to be 2.5$\lambda _B$. Finally, measurement guidelines are provided to assess the accuracy of 3D SDR measurements.

Graphical Abstract

Appendices

Appendix 1

1.1 Resolution error curve

Fig. 16

Effect of spatial resolution on mean 1D SDR based on the model spectrum for Re$_\lambda = 164$

Figure 16 shows the effect of resolution on one-dimensional mean scalar dissipation rate. This curve is obtained using the model dissipation spectrum from Pope (2000) in an identical manner as of Wang et al. (2007b). The one-dimensional dissipation spectrum is filtered using the box, Gaussian, and sharp-spectral filters. The filter expressions can be found in Pope (2000) or Wang et al. (2007b). The mean SDR obtained from the unfiltered spectrum is denoted as $\langle \chi _1\rangle$, whereas its filtered counterpart is denoted as $\langle \chi _{1f}\rangle$. $\varDelta _H$ is the characteristic filter width, and $\lambda _B$ is the Bachelor length scale which is taken as $60~\upmu$m based on the dissipation spectrum presented in Fig. 6b. A crude estimate of the Taylor scale Reynolds number (Re$_\lambda$) is obtained by considering the longitudinal integral length scale, $L_{11} = 0.3$Da, where Da is the inner diameter of the outer tube. Next, the length scale used to evaluate the model spectrum is assumed to be $L = 0.5L_{11}$. Although the Re$_\lambda$ (and corresponding spectrum) is uncertain due to the assumed scale, the resolution curve for SDR is not strongly sensitive to the Re$_\lambda$ as shown by Wang et al. (2007b). Therefore, the above results can be used to obtain the resolution requirement for the present flow.

For the sharp-spectral filter, SDR can be obtained accurately even with the resolution of $3\lambda _B$, whereas for the box or Gaussian filtered data, the error at $3\lambda _B$ is $\approx 7\%$. For the resolution of $1\lambda _B$, the SDR accuracy of better than $1\%$ can be achieved even with the box or Gaussian filtered data. Note that these findings are deduced from the noise-free modeled spectrum. For measurements, in addition to noise, the differencing stencil also influences the resolution requirement. Relative to the 2-point difference stencil, the resolution requirement for the 3-point central difference stencil is stringent (Wang et al. 2007b).

Appendix 2

1.1 Mean 2D SDR

See Fig. 17.

Fig. 17

Mean 2D SDR at mean $\varDelta s = 109~\upmu$m: a from I1 field of C1/L1 data at 9 mJ, and b from I2 field of C2/L2 data at 36 mJ, and c difference between SDR from I2 and I1

Appendix 3

1.1 Convergence

See Fig. 18.

Fig. 18

Convergence of the ensemble- and area-averaged normalized $\chi _{3C}$ with the number of samples: a 52 samples with instantaneous $\varDelta s = 143~\upmu$m, and b 165 samples with mean $\varDelta s = 140~\upmu$m where instantaneous $\varDelta s$ ranges from 135 to 165 $\upmu$m

Appendix 4

1.1 Energy and dissipation spectra

See Figs. 19 and 20.

Fig. 19

Mean spectrum of fluctuating scalar, $\zeta ^\prime$, for the mid-ROI a radial energy, b radial dissipation, c axial energy, d axial dissipation. The Bachelor scale is indicated for each PSD curve

Fig. 20

Mean spectrum of fluctuating scalar, $\zeta ^\prime$, for the top ROI a radial energy, b radial dissipation, c axial energy, d axial dissipation. The Bachelor scale is indicated for each PSD curve