Statistically advanced, self-similar, radial probability density functions of atmospheric and under-expanded hydrogen jets

Abstract

This paper presents improved statistical insight regarding the self-similar scalar mixing process of atmospheric hydrogen jets and the downstream region of under-expanded hydrogen jets. Quantitative planar laser Rayleigh scattering imaging is used to probe both jets. The self-similarity of statistical moments up to the sixth order (beyond the literature established second order) is documented in both cases. This is achieved using a novel self-similar normalization method that facilitated a degree of statistical convergence that is typically limited to continuous, point-based measurements. This demonstrates that image-based measurements of a limited number of samples can be used for self-similar scalar mixing studies. Both jets exhibit the same radial trends of these moments demonstrating that advanced atmospheric self-similarity can be applied in the analysis of under-expanded jets. Self-similar histograms away from the centerline are shown to be the combination of two distributions. The first is attributed to turbulent mixing. The second, a symmetric Poisson-type distribution centered on zero mass fraction, progressively becomes the dominant and eventually sole distribution at the edge of the jet. This distribution is attributed to shot noise-affected pure air measurements, rather than a diffusive superlayer at the jet boundary. This conclusion is reached after a rigorous measurement uncertainty analysis and inspection of pure air data collected with each hydrogen data set. A threshold based upon the measurement noise analysis is used to separate the turbulent and pure air data, and thusly estimate intermittency. Beta-distributions (four parameters) are used to accurately represent the turbulent distribution moments. This combination of measured intermittency and four-parameter beta-distributions constitutes a new, simple approach to model scalar mixing. Comparisons between global moments from the data and moments calculated using the proposed model show excellent agreement. This was attributed to the high quality of the measurements which reduced the width of the correctly identified, noise-affected pure air distribution, with respect to the turbulent mixing distribution. The ignitability of the atmospheric jet is determined using the flammability factor calculated from both kernel density estimated (KDE) PDFs and PDFs generated using the newly proposed model. Agreement between contours from both approaches is excellent. Ignitability of the under-expanded jet is also calculated using KDE PDFs. Contours are compared with those calculated by applying the atmospheric model to the under-expanded jet. Once again, agreement is excellent. This work demonstrates that self-similar scalar mixing statistics and ignitability of atmospheric jets can be accurately described by the proposed model. This description can be applied with confidence to under-expanded jets, which are more realistic of leak and fuel injection scenarios.

Appendix

The centerline self-similarity of the atmospheric hydrogen jet and the under-expanded hydrogen jet has been established. This appendix documents the conventional radial collapse of the mean, rms, skewness, and kurtosis statistics for the atmospheric jet. The collapse of these statistics for the under-expanded jet has previously been published (Ruggles and Ekoto 2012).

Figure 21 shows the Gaussian fit of Richards and Pitts (1993) compared to the present collapsed data. The agreement for the mean is excellent. The rms collapse is similar to the second-moment collapse, in that the values converge to the rms values of pure air at η = 0.3. The fourth-order polynomial from Richards and Pitts (1993) does not extend beyond η = 0.25 and is unable to converge toward pure air values. The data and polynomial agree very well at positions where η < 0.15, after which the polynomial and data begin to deviate. This is attributed to two factors; the first is the difference in diagnostic noise characteristics of the two studies. As intermittency falls from unity, the presence of the pure air distribution increases. As has been documented, the pure air distribution width (rms) is dictated by the associated shot noise of the diagnostic. Therefore, at positions where more pure air measurements are made, the global rms value starts to represent the rms of pure air measurements of the specific diagnostic. Secondly, the treatment of negative pure air samples is unknown in all literature studies. In this work, negative values have been included. If they are omitted or converted to absolute values, the corresponding global rms values of the pure air distribution would be reduced. Skewness and kurtosis are not normalized with respect to self-similarity (Eq. 7). As a result beyond radial positions, η = 0.15, the collapse of values rapidly deteriorates to a completely random state at the edge of the jet. This shows that plotting skewness and kurtosis against η is insufficient to investigate the self-similarity of higher-order moments.

Fig. 21

Conventional self-similar collapse of mean, rms, skewness, and kurtosis statistics with the literature curve fits (Richards and Pitts 1993)

Flammability factor images for both jets, extending to the far downstream, were generated using the non-dimensional scalar KDE PDFs. These are then rescaled to mass fraction using the appropriate mean centerline mass fraction value, $\bar{Y}_{\text{CL}}$. A further rescaling is required to convert the PDFs to mole fraction. Once completed, integration between the flammable limits yields the flammability factor. For a given axial position, the flammability factor values for all η positions are calculated. Hermite interpolation is performed between calculated values where necessary (Fig. 22).

Fig. 22

Flammability factor images determined using the KDE PDF method

Table 3 documents the required moments to generate the turbulent four-parameter beta-distributions and corresponding intermittency values. Once generated for the non-dimensional scalar, the distributions are rescaled to mass fraction by using the appropriate mean mass fraction centerline value, $\bar{Y}_{\text{CL}}$. Using the appropriate intermittency value, global scalar mixing moments and flammability factor (once rescaled to mole fraction) values can be determined at any axial and radial position.

Table 3 Table of the first to fourth turbulent moments and intermittency

The following details the functional definition of the four-parameter beta-distribution, corresponding moments, shape parameters (p, q), and limits (a, b) (Johnson and Kotz 1970).

$${\rm B}\left( {x;a,b,p,q} \right) = \frac{{\varGamma \left( {p + q} \right)}}{\varGamma \left( p \right)\varGamma \left( q \right)}\frac{{\left( {x - a} \right)^{p - 1} \left( {b - x} \right)^{q - 1} }}{{\left( {b - a} \right)^{p + q - 1} }}\quad \left( {a \le x \le b;p,q \ge 0} \right)$$

(10)

Moments:

$${\text{Mean}} = a + \left( {b - a} \right)p/\left( {p + q} \right)$$

(11)

$$M2 = \left( {b - a} \right)^{2} pq/\left[ {\left( {p + q} \right)^{2} \left( {p + q + 1} \right)} \right]$$

(12)

$$M3 = \frac{{2\left( {q - p} \right)\sqrt {p^{ - 1} + q^{ - 1} + \left( {pq} \right)^{ - 1} } }}{p + q + 2}M2^{3/2}$$

(13)

$$M4 = \frac{{3\left( {p + q + 1} \right)\left[ {2\left( {p + q} \right)^{2} + pq\left( {p + q - 6} \right)} \right]}}{{pq\left( {p + q + 2} \right)\left( {p + q + 3} \right)}}M2^{2}$$

(14)

Parameters:

$$p,q = \frac{r}{2}\left\{ {1 \pm \left( {r + 2} \right)\frac{{\sqrt {\left( {M3/M2^{3/2} } \right)} }}{\sqrt D }} \right\}$$

(15)

$$b - a = \sqrt {M2} \sqrt D$$

(16)

$$a = mean - \left( {b - a} \right)p/\left( {p + q} \right)$$

(17)

where

$$r = \frac{{6\left( {\left( {M4/M2^{2} } \right)^{2} \left( {M3/M2^{3/2} } \right)^{2} - 1} \right)}}{{6 + 3\left( {M3/M2^{3/2} } \right)^{2} - 2\left( {M4/M2^{2} } \right)^{2} }}\quad D = \left( {r + 2} \right)^{2} \left( {M3/M2^{3/2} } \right)^{2} + 16\left( {r + 1} \right)$$

(18)