Comparisons are made between the Advection-Diffusion Equation (ADE) approach for particle transport and the two fluid model approach based on the PDF method. In principal the ADE approach offers a simpler way of calculating the inertial deposition of particles in a turbulent boundary layer than that based on the PDF approach. However the ADE equations that have recently been used are only strictly valid for a simple Gaussian process when particle inertia is small. Using a prescribed but in general non-Gaussian random particle velocity field, it is shown that the net particle mass flux contains an extra drift term to that from the mean velocity of the particle velocity field, associated with the compressibility of the velocity field. Furthermore the diffusive flux in general depends not only upon the gradient of the mean concentration (true only for a Gaussian random flow field) but also upon higher order derivatives whose relative contribution depends on diffusion coefficients Dijk... etc. These coefficients depend upon the statistical moments associated with random displacements and compressibility of the particle flow field along particle trajectories which in turn depend upon particle inertia. In contrast the PDF approach offers the advantage of using a simple gradient (Gaussian) approximation in particle phase space which can lead to a non-Gaussian spatial dispersion process when particle inertia is important. Conditions based on the particle mean free path are derived for which a simple ADE is appropriate. Some of the features of particle transport in an inhomogeneous turbulent flow are illustrated by examining particle dispersion in a random flow field composed of pairs of counter rotating vortices which has an rms velocity which increase linearly from a stagnation point.
Comparison of Recent Model Equations for Particle Deposition in a Turbulent Boundary Layer With Those Based on the PDF Approach
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Reeks, MW. "Comparison of Recent Model Equations for Particle Deposition in a Turbulent Boundary Layer With Those Based on the PDF Approach." 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. 469-481. ASME. https://doi.org/10.1115/FEDSM2003-45734
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