Probability density function turbulent diffusion flame

Drake, M. C., R. W. Pitz, and W. Shyy (1986). Conserved scalar probability density functions in a turbulent jet diffusion flame. Journal of Fluid Mechanics 171, 27-51. 

Figure 4. Probability density functions of temperature for Ht-air turbulent diffusion flame determined at various radial positions 134 mm downstream of the fuel line tip according to procedures indicated in Figure 3. The measurement positions are drawn schematically in the center of the figure to correspond to the radial positions r on the scale at the RHS.

Figure 5. Probability density function (pdf or histogram) for temperature X velocity for turbulent diffusion flame. These data correspond to a test zone along the axis, 50 fuel-tip diameters downstream from the fuel line tip.

Because of these difficulties with moment methods for reacting flows, we shall not present them here. A number of reviews are available [22], [25], [27], [32]. There are classes of turbulent combustion problems for which moment methods are reasonably well justified [40]. Since the computational difficulties in use of moment methods tend to be less severe than those for many other techniques (for example, techniques involving evolution equations for probability-density functions), they currently are being applied to turbulent combustion in relatively complex geometrical configurations [22], [31], [32]. Many of the aspects of moment methods play important roles in other approaches, notably in those for turbulent diffusion flames (Section 10.2). We shall develop those aspects later, as they are needed. 

For initially nonpremixed reactants, two limiting cases may be visualized, namely, the limit in which the chemistry is rapid compared with the fluid mechanics and the limit in which it is slow. In the slow-chemistry limit, extensive turbulent mixing may occur prior to chemical reaction, and situations approaching those in well-stirred reactors (see Section 4.1) may develop. There are particular slow-chemistry problems for which the previously identified moment methods and age methods are well suited. These methods are not appropriate for fast-chemistry problems. The primary combustion reactions in ordinary turbulent diffusion flames encountered in the laboratory and in industry appear to lie closer to the fast-chemistry limit. Methods for analyzing turbulent diffusion flames with fast chemistry have been developed recently [15], [20], [27]. These methods, which involve approximations of probability-density functions using moments, will be discussed in this section. 

Distributions like those in Figure 10.4, for example, indicate that Yp or T differs from Yp(Z) or T(Z), respectively. If mixing were complete in the sense that all probability-density functions were delta functions and fluctuations vanished, then differences like T — T Z) would be zero. That this situation is not achieved in turbulent diffusion flames has been described qualitatively by the term unmixedness [7]. Although different quantitative definitions of unmixedness have been employed by different authors, in one way or another they all are measures of quantities such as Yp — Yp(Z) or T — T(Z). The unmixedness is readily calculable from P(Z), given any specific definition (see Bilger s contribution to [27]). 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

C. Bonniot, R. Borghi, Joint probability density function in turbulent combustion, Acta Astronautica, vol. 6, pp. 309-327 (1979). Janicka, W. Kollmann, A Two-variables formalism for the treatment of chemical reactions in turbulent H -Air diffusion flame, 17 Symp. (Int.) on Combustion, pp. 421-430, The Comb. Inst., Pittsburgh (1979). 

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