Flame probability density functions

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. 

Raju, M. S. (1996). Application of scalar Monte Carlo probability density function method for turbulent spray flames. Numerical Heat Transfer, Part A 30, 753-777. 

The figure shows U >. S L in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for very intense turbulence, U > SL. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unbumed mixture near the burned gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more importantly, one that is not bimodal. 

The present study is to elaborate on the computational approaches to explore flame stabilization techniques in subsonic ramjets, and to control combustion both passively and actively. The primary focus is on statistical models of turbulent combustion, in particular, the Presumed Probability Density Function (PPDF) method and the Pressure-Coupled Joint Velocity-Scalar Probability Density Function (PC JVS PDF) method [23, 24]. 

The Presumed Probability Density Function method is developed and implemented to study turbulent flame stabilization and combustion control in subsonic combustors with flame holders. The method considers turbulence-chemistry interaction, multiple thermo-chemical variables, variable pressure, near-wall effects, and provides the efficient research tool for studying flame stabilization and blow-off in practical ramjet burners. Nonreflecting multidimensional boundary conditions at open boundaries are derived, and implemented into the current research. The boundary conditions provide transparency to acoustic waves generated in bluff-body stabilized combustion zones, thus avoiding numerically induced oscillations and instabilities. It is shown that predicted flow patterns in a combustor are essentially affected by the boundary conditions. The derived nonreflecting boundary conditions provide the solutions corresponding to experimental findings. 

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.

Probability density functions, or histograms, of the product of instantaneous temperature x velocity were obtained through use of this combustion probe system for a variety of downsteam and radial flame test positions. A typical histogram is shown in Fig. 3, while Fig. 4 displays the same data (as well as data for a test position further downsteam) in a "scattergram" format i.e., in a plot of velocity vs. temperature. Here, each datum corresponds to a specific shot, while the histogram bins correspond to integrated results from numbers of shots. 

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]). 

FIGURE 10.4. Illustration of probability-density functions for the mixture fraction, fuel mass fraction, oxidizer mass fraction, and temperature for a jet-type diffusion flame in the flame-sheet approximation. 

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. 

---