(376e) Solution of the Fokker-Planck Equation Using the Extended Quadrature Method of Moments

One of the important attractions of employing the transported probability density function (PDF) methods for turbulent reacting flows is the fact that the chemical source terms are treated exactly. The composition probability density function (PDF) evolves with convective transport in physical space due to the mean velocity (macromixing), the turbulent diffusivity which transports the composition PDF in physical space (mesomixing), the transport in composition space due to molecular mixing (micromixing) and chemical reactions. The key model term of our interest in this study in the transport PDF equation is the molecular mixing term, which describes how molecular diffusion affects both the shape of the PDF and the rate of scalar variance decay. Molecular mixing is expressed using the Fokker-Planck model (Fox, 2003, 1994), which is an extension of the interaction-by-exchange-with-the-mean (IEM) model. The IEM model, widely used in chemical-reaction engineering and computational combustion due to its simple form, assumes the linear relaxation of the scalar towards its mean, while the Fokker-Plank model considers, in addition, the effect of the differential diffusion process to mimic the mixing. Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. An extended quadrature-based moment method (EQMOM) (Yuan et al., 2012; Chalons C. Fox R.-O. Massot M., 2010) is used to close the transport equation for the composition PDF. The β kernel density function is used for EQMOM since the scalar composition is represented by a scalar bounded between -1 and 1. The PDF predicted by the EQMOM model in the same conditions studied in the direct numerical simulation of Eswaran and Pope (1988) are reported and compared to the DNS predictions.

References

Chalons, C., Fox, R. O., Massot, M., 2010. A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework. Studying Turbulence Using Numerical Simulation Databases, Center for Turbulence Research, Summer Program 2010, Stanford University 347–358.

Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids 31, 506–520.

Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press.

Fox, R.O., 1994. Improved Fokker–Planck model for the joint scalar, scalar gradient PDF. Physics of Fluids 6, 334–348.

Yuan, C., Laurent, F., Fox, R.O., 2012. An extended quadrature method of moments for population balance equations. Journal of Aerosol Science 51, 1–23.