Evolution equations for the joint probability of several compositions in turbulent combustion

Conference · Fri Jan 01 00:00:00 EST 2010

OSTI ID:1014450

Bakosi, Jozsef ^[1]

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1. Los Alamos National Laboratory

One-point statistical simulations of turbulent combustion require models to represent the molecular mixing of species mass fractions, which then determine the reaction rates. For multi-species mixing the Dirichlet distribution has been used to characterize the assumed joint probability density function (PDF) of several scalars, parametrized by solving modeled evolution equations for their means and the sum of their variances. The PDF is then used to represent the mixing state and to obtain the chemical reactions source terms in moment closures or large eddy simulation. We extend the Dirichlet PDF approach to transported PDF methods by developing its governing stochastic differential equation (SDE). The transport equation, as opposed to parametrizing the assumed PDF, enables (1) the direct numerical computation of the joint PDF (and therefore the mixing model to directly account for the flow dynamics (e.g. reaction) on the shape of the evolving PDF), and (2) the individual specification of the mixing timescales of each species. From the SDE, systems of equations are derived that govern the first two moments, based on which constraints are established that provide consistency conditions for material mixing. A SDE whose solution is the generalized Dirichlet PDF is also developed and some of its properties from the viewpoint of material mixing are investigated. The generalized Dirichlet distribution has the following advantages over the standard Dirichlet distribution due to its more general covariance structure: (1) its ability to represent differential diffusion (i.e. skewness) without affecting the scalar means, and (2) it can represent both negatively and positively correlated scalars. The resulting development is a useful representation of the joint PDF of inert or reactive scalars in turbulent flows: (1) In moment closures, the mixing physics can be consistently represented by one underlying modeling principle, the Dirichlet or the generalized Dirichlet PDF, and (2) based on the SDEs transported PDF mixing models for multi-species diffusion can be constructed by specifying the SDE coefficients.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: AC52-06NA25396
• OSTI ID: 1014450
• Report Number(s): LA-UR-10-03441; LA-UR-10-3441; TRN: US201110%%908
• Resource Relation: Conference: 49th AIAA Aerospace Sciences Meeting and Aerospace Exposition ; January 4, 2011 ; Orlando, FL
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ASYMMETRY
CHEMICAL REACTIONS
CLOSURES
COMBUSTION
DIFFERENTIAL EQUATIONS
DIFFUSION
DISTRIBUTION
LARGE-EDDY SIMULATION
PHYSICS
PROBABILITY
PROBABILITY DENSITY FUNCTIONS
REACTION KINETICS
SCALARS
SHAPE
SIMULATION
SOURCE TERMS
SPECIFICATIONS
STATISTICS
TRANSPORT
TURBULENT FLOW