Scalar Probability Density Function Research Articles

A Generalized Conditional Scalar Dissipation-Mixture Fraction Joint PDF for Flamelet Modeling of Non-Premixed Flames

Abstract One problem in flamelet modeling of turbulent non-premixed flames is that of formulating a scalar dissipation rate-mixture fraction joint probability density function (pdf). While the individual flamelet calculation provides a conditional scalar dissipation rate, the turbulent flame calculation provides a marginal scalar dissipation rate. Thus, linking the laminar and turbulent flame calculations through the joint pdf is not straightforward, and, for simplicity, it is often assumed that the marginal and conditional probability densities are equal. The present paper assesses the validity of this assumption and examines its consequences in a numerical example. A conditional, scalar dissipation rate-mixture fraction joint pdf is developed which is based upon numerical solutions of the laminar counterflow diffusion flame and functional relationships among the scalar dissipation rate, the conditional scalar dissipation rate, and the mixture fraction. Interestingly, the result, when applied to a turbulent non-premixed flame, results in predictions of NO concentration which are not significantly different. The analysis also suggests experiments which may provide insight into flamelet structure, thereby assessing the validity of the idealized, underlying laminar counterflow configuration used in the flamelet modeling of non-premixed flames.

Monte Carlo PDF method for turbulent reacting flow in a jet-stirred reactor

A stochastic algorithm for the solution of the modelled scalar probability density function (PDF) transport equation for single-phase turbulent reacting flow is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles of N representative values of the thermochemical variables in each cell of a non-uniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor.

Mapping closure and non-Gaussianity of the scalar probability density functions in isotropic turbulence

An attempt is made to assess the amplitude mapping closure for turbulent scalar fields. The closure assumption is shown to be in very good agreement with the results obtained from direct numerical simulations (DNS) of passive scalars in isotropic turbulence. The amplitude mapping closure is then employed to investigate the non-Gaussian properties of one-point scalar probability density functions (pdf) in isotropic flows. It is shown that the general relaxation of a scalar pdf to a Gaussian distribution can only be expected when the scalar variance is very small. Some non-Gaussian characteristics imposed by initial and boundary conditions persist throughout the course of scalar evolution.

An analytical solution for the scalar probability density function in homogeneous turbulence

The probability density function (pdf) of a passive scalar in homogeneous turbulence is studied using the mapping closure model recently developed by Kraichnan and co-workers. An analytical solution for the mapping equation is derived. This solution preserves the boundedness of the scalar fields and yields the pdf’s that relax to Gaussian distributions, in agreement with experiments and numerical simulations.

A new micromixing model for turbulent reactors

A model based on the theory of instantaneous reactions is proposed which allows quantification of turbulent micromixing in terms of the scalar probability density function (PDF) and its variance. On a unified reactive mixing plot, the asymptotes for complete mixing (when each of the reactants is limiting) form an upper bound and are linear. Knowledge of these two theoretical asymptotes is fruitfully exploited using the Churchill-Usagi correlational approach to obtain a two-parameter equation for the reactive mixing curves. This analytical expression can be differentiated once to obtain the cumulative probability distribution, differentiated twice to determine the PDF itself, or integrated once to obtain the variance. The exponential parameter in the Churchill-Usagi correlation, which is a measure of the deviation from the asymptotic behavior (of complete mixing, in this case), is directly related to the variance. It varies from zero to negative infinity, and provides a practical measure of the entire range of micromixing behavior from complete segregation to complete mixing. The model reduces experimentation significantly; in principle, it requires only a single non-reacting experiment to obtain the macromixing parameter and a single reacting experiment to obtain the micromixing parameter. The model is validated using previously published experimental data (Vassilatos, 1964; Mao, 1969) and results in PDF shapes more reliable than those obtained earlier (Shenoy and Toor, 1989).

Simulations of scalar mixing in grid turbulence using an eddy-damped closure model

A physical-space eddy-damped quasinormal Markovian model for the two-point scalar correlation function is used to simulate wind tunnel experiments on scalar mixing. The simulations closely approximate experimentally observed trends of scalar variance and length scales in both heated-grid and heated-screen experiments. The simplicity of the model should enable its effective use in modeling the analogous turbulent mixing terms in the two-point scalar probability density function equations.

Conditional probability density functions measured in turbulent nonpremixed flames of methane near extinction

Spontaneous Raman-Rayleigh measurements of temperature and species concentrations have been made in turbulent nonpremixed flames of methane close to extinction. For each data point, five different values of reactedness are calculated from the measured temperature and the mass fractions of CH 4, O 2, H 2O and CO 2. These data are then sampled with respect to five ranges of mixture fraction and the conditional single and joint probability density functions (PDFs) of reactedness calculated from the various reactive scalars are presented for each range. Conditional means and root mean squares (rms) of fluctuations of reactedness, temperature, and species mass fractions are also presented. It is found that for mixture fraction ranges lying below the rich reaction limit, ξ < ξ R ( ξ R = 0.087 for methane) combustion is generally bimodal and fluid samples are either fully burned or unburned. For ranges with ξ > ξ R, the majority of mixtures are partially reacted and the conditional PDFs have centered distributions. The correlation between the conditional PDFs of various reactive scalars indicate that more than a single-step reaction mechanism is required to adequately describe the chemistry. Close to extinction, the flame was kept alight by fully burned pockets of fluid that provide a source of reignition for mixtures with slower mixing rates. The flame blows off when the frequency of occurrence of these pockets becomes small.

Conserved scalar probability density functions in a turbulent jet diffusion flame

The first four moments of conserved scalar probability density functions (p.d.f.'s) measured by Raman scattering in an H2 turbulent jet diffusion flame are analysed and compared with those found by Pitts &amp; Kashiwagi (1984) in a non-reacting CH4 jet. The measurements are in good agreement, indicating that heat release and combustion have little effect on p.d.f. shapes. However, the measured p.d.f.'s are not qualitatively similar to the simple forms often assumed in combustion modelling. A three-zone model by Effelsberg &amp; Peters was used to separate the experimental p.d.f.'s into a delta function (non-turbulent zone), a Gaussian (turbulent zone) and the remainder (interface zone). The interface zone contributed as much as 90% of the total p.d.f. in both the H2 flame and the non-reacting CH4 jet. A physical interpretation for the existence of broad interface zones in reacting and non-reacting turbulent jet flows is suggested based upon large-scale structures.

A composite model for the conserved scalar pdf

A parametric expression for the conserved scalar probability density function (pdf) is derived on the basis of a physical interpretation of the turbulent signal. The scalar signal is split into three parts: a fully turbulent part, a superlayer part, and a outer flow part. Each part is represented by a different form of the pdf. The composite pdf which is obtained by adding these parts together has four parameters in addition to the intermittency factor. It can be integrated so that algebraic relations between these parameters and the first four moments are obtained. Application of the formulation to LaRue's and Libby's measurements in the plane turbulent wake indicates the existence of transitions that are interpreted as internal superlayers far inside the turbulent flow.

Statistical Constraints on Scalar Correlations

Models for the scalar probability density function (pdf) have to be developed to achieve closure of turbulent transport equations for mixing and reacting flows. The best statistical bounds on a number of moments for tvvoand three-species flows have been derived and used to construct and test a delta function typical eddy pdf model. If has been proven that for two species a rational pdf composed only of delta functions can always be constructed at any point within the statistically valid moment space. The delta function model and a canonical pdf model have been directly compared to experimental pdf measurements and both models show good agreement for higher-order moments, but the delta function model is simpler to construct and is recommended for closure of the transport equations for mixing flows.

Radioactivity of archeological specimens

A parametric expression for the conserved scalar probability density function (pdf) is derived on the basis of a physical interpretation of the turbulent signal. The scalar signal is split into three parts: a fully turbulent part, a superlayer part, and a outer flow part. Each part is represented by a different form of the pdf. The composite pdf which is obtained by adding these parts together has four parameters in addition to the intermittency factor. It can be integrated so that algebraic relations between these parameters and the first four moments are obtained. Application of the formulation to LaRue's and Libby's measurements in the plane turbulent wake indicates the existence of transitions that are interpreted as internal superlayers far inside the turbulent flow.