The paper addresses an apparent incompatibility between the two current formulations of the PDF approach for the dispersion of particles in a uniform shear flow. It is shown that this incompatibility arises through neglect of the inertial convection term in the transport equation for the mean carrier flow velocity local to a particle. Evaluating this term for a Gaussian process gives identical results for both formulations. This also resolves a long standing incompatibility in previous forms for the fluid point diffusions coefficients in a simple shear. A closure approximation is given based on a Gaussian model for the contribution due to the fluctuating shear which has previously been assumed to be white noise in the generalised Langevin model (GLM) due to Simonin, Deutsch and Minier.
Closure Approximations for PDF Equations Based on the Simonin-Deutsch-Minier (SDM) Equation for Gas-Particle Flows
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Reeks, MW. "Closure Approximations for PDF Equations Based on the Simonin-Deutsch-Minier (SDM) Equation for Gas-Particle Flows." Proceedings of the ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference. Volume 1: Fora, Parts A, B, C, and D. Honolulu, Hawaii, USA. July 6–10, 2003. pp. 461-467. ASME. https://doi.org/10.1115/FEDSM2003-45733
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