Micro-PDF methods for turbulent flow and reactions

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

The one-point joint velocity-composition PDF transport equation expresses the evolution of the PDF by transport in real space, in velocity space, and in composition space and can be derived from the Navier-Stokes equations 

The PDF transport equation (12.4.2-2) is deterministic. Convection appears in closed form. The conditional acceleration term (12.4.2-3) and the conditional diffusion / reaction term (12.4.2-4) are unclosed. In the conditional acceleration term (12.4.2-3), only the first term in the right hand side is actually unclosed. The effects of gravity are dealt with in an exact way. In the conditional diffusion / reaction term (12.4.2-4), the turbulent mixing term is unclosed, but the reaction rate appears in closed form whatever the complexity and nonlinearity of the reaction kinetics. The remaining part of this section focuses on closure models. 

Applying this decomposition, the effects of the mean pressure gradient and the mean shear stress also appear in closed form in the PDF transport equation (12.4.2-2). 

At high Reynolds numbers, the fluctuating pressure field term dominates the fluctuating shear stress term in (12.4.2-6). It results from interactions between the turbulence and the mean velocity gradients, the so-called rapid-pressure contribution, and from the turbulence proper, the so-called slow-pressure contribution. To simulate the rapid-pressure related process. Pope [1985] proposed a linear deterministic model  

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Micro-PDF Methods for Turbulent Flow and Reactions Micro-PDF Moment Methods Computational Fluid Dynamics... 

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