Probability Density Function Evolution Equation Research Articles

On the transport equation for probability density functions of turbulent vorticity fields.

Vorticity random fields of turbulent flows (modelled over the vorticity equation with random initial data for example) are singled out as the main dynamic variables for the description of turbulence, and the evolution equation of the probability density function (PDF) of the vorticity field has been obtained. This PDF evolution equation is a mixed type partial differential equation (PDE) of second order which depends only on the conditional mean (which is a first-order statistics) of the underlying turbulent flow. This is in contrast with Reynolds mean flow equation which relies on a quadratic statistics. The PDF PDE may provide new closure schemes based on the first-order conditional statistics, and some of them will be described in the paper. We should mention that the PDF equation is interesting by its own and is worthy of study as a PDE of second order.

Modeling return to isotropy using kinetic equations

Kinetic equations modeling the behavior of the velocity probability density function (PDF) in homogeneous anisotropic decaying turbulence are hypothesized and their implications for return-to-isotropy are investigated. Anisotropic turbulent decay is a parametrically simple but theoretically complex turbulent flow that is dominated by nonlinear interactions. The physical implications of the Bhatnagar–Gross–Krook model, a relaxation model, and the Fokker–Planck model for the “collision” term in the PDF evolution equation are analyzed in detail. Using fairly general assumptions about the physics, three different parameter-free return-to-isotropy models are proposed. These models are compared with experimental data, classical models, and analytical limits. The final model expression is particularly interesting, and can easily be implemented in classic Reynolds stress transport models.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Assumed PDF turbulence-chemistry closure with temperature-composition correlations

An assumed probability density function (PDF) approach that accounts for temperature and composition fluctuations, including effects of temperature-composition correlations, has been developed. Parametric studies were performed to determine the gross behavior of the assumed PDF and provide insight into the expected trends of the model for a given kinetic mechanism. The model was implemented into an existing Navier-Stokes solver and applied to a supersonic, co-axial H 2/air burner. Proper prediction of the turbulence effects on the chain initiating step of the kinetic process is a first-order concern in supersonic combustion applications because of small flow residence times. Temperature-composition correlations were extracted and compared with results obtained by direct integration of a PDF evolution equation. The results of this comparison showed that the model properly predicted both the magnitude and sign of the cross-correlation terms associated with the chain initiating step of the H 2/air kinetic mechanism. Correlations for all but one of the remaining kinetic steps qualitatively reproduced those obtained by direct integration.

Assumed and evolution probability density functions in supersonic turbulent combustion calculations

The objective of this investigation is to compare the use of assumed probability density function (PDF) approaches for modeling supersonic turbulent reacting flowfields with the more elaborate approach where the PDF evolution equation is solved. Previous calculations using assumed PDFs have shown modest improvements in the prediction of mean flow variables when compared with experimental data. However, these PDFs were unable to predict the higher order correlations with any reasonable degree of accuracy. Solving the evolution equation for the PDF did show slight improvements in these correlations when compared with experiment, but at a substantial cost in computer storage and time. Both approaches yielded comparable mean flow quantities.

A closed probability density function evolution equation for stochastic diffusion

A closed probability density function (PDF) evolution equation is obtained for a homogeneous scalar field undergoing stochastic diffusion. The closure assumes that the scalar field is made up of an ensemble of one-dimensional periodic waves.

Probability density function approach for compressible turbulent reacting flows

The objective of the present work is to extend the probability density function (PDF) tubulence model to compressible reacting flows. The proability density function of the species mass fractions and enthalpy are obtained by solving a PDF evolution equation using a Monte Carlo scheme. The PDF solution procedure is coupled with a compression finite-volume flow solver which provides the velocity and pressure fields. A modeled PDF equation for compressible flows, capable of treating flows with shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed. Two super sonic diffusion flames are studied using the proposed PDF model and the results are compared with experimental data; marked improvements over solutions without PDF are observed.