Development of the PDF kinetic approach for modelling inertial particle dispersion in turbulent boundary layers

Abstract

This thesis presents developments in the use of probability density function (PDF) kinetic equations to model the dispersion of inertial particles in turbulent boundary layers. The PDF kinetic equation is used as a master equation from which to construct continuum equations for the particle-phase, and these continuum equations form an in nite set of coupled equations which require closure in the particle velocity statistics. Furthermore, the continuum equations contain dispersion tensors which describe the e ect of the underlying uid turbulence on the dispersion of the particles throughout the ow eld. These dispersion tensors themselves require closure and in this thesis new closure models are developed which are non-local and attempt to take into account the e ects of turbulence inhomogeneity, anisotropy and particle-wall collisions on the dispersion tensors. The rst closure model developed is for particles dispersing under Stokes drag forcing only; appropriate for particles whose material density is much greater than that of the uid in which they are dispersed. This closure model is tested against equivalent particle tracking simulation data over a range of particle sizes and the closure model predictions are found to be in excellent agreement. In contrast to the new closure model predictions, the traditional `local' approximations to the dispersion tensors are found to be in signi cant error when compared to the particle tracking data. The closure model is then developed to account for particles dispersing under Stokes drag, added mass and gravitational forcing; added mass forcing being important for particles whose material density is comparable to or less than that of the uid in which they are dispersed. The modelling is presented and a discussion is given regarding the various complex terms that require approximation in this closure model. The closure model predictions are then compared against the alternative local approximations. It is seen that with added mass forcing the local approximations can be qualitatively and quantitatively di erent to the non-local predictions, whereas under only a drag force, errors in the local approximations are mainly quantitative. Finally, consideration is given to the forms of the dispersion tensors appearing in the PDF and continuum equations. It is shown theoretically that the dispersion tensors (and therefore the PDF and continuum equations themselves) are free from the so called `spurious drift' phenomena associated with certain types of models for predicting the dispersion of uid particles in incompressible, inhomogeneous turbulent ows. However, it is also shown that closure approximations applied to the dispersion tensors may result in the introduction of a spurious drift. Nevertheless, it is demonstrated that the arti cial drift introduced by closure approximations does not have any appreciable a ect on the dispersion tensors when they are describing the dispersion of inertial particles.