Developing Novel Applications of PDF Based Models for Studying Turbulent Disperse Flows

Dispersed flows of droplets and particles abound in nature; from clouds, mist and fogs to the long-range transport of fine dust released in desert storms or in volcanic eruptions. Such flows control the weather and influence the climate. They play key roles in many industrial energy processes from spray drying, pneumatic and slurry transport, fluidized beds, to coal gasification and mixing and combustion processes. They can have a profound effect on our health and quality of life, from the spread of communicable diseases to the inhalation of very fine air-borne particulates. Modelling and computing all of the above process accurately and efficiently is therefore important in our control of the environment, improving our health, and in the design and improvement of industrial processes.This proposal is about developing and applying a novel approach to modelling and computing these processes. The approach is called the probability density function (pdf) approach and is analogous to the kinetic theory of gases in that, central to the approach, is a master equation (analogous to the Maxwell-Boltzmann equation) from which can be derived the basic continuum conservation equations and associated constitutive relations for the treatment of the particulate phase as a fluid (e.g.expressions for the viscous shear stress of the fluid in terms of the rate of straining of the fluid). Whilst two-fluid model equations (for the treatment of both the carrier and particle phases as fluids) are the most efficient way of computing the transport of both phases, and ideal for implementation within the standard framework of a CFD code, it has long been recognised that they have severe limitations because the wall-boundary conditions necessary for solution of the equations for the dispersed phase are incompatible with the natural wall boundary conditions. This is because particle wall interactions are concerned with changes in the particle velocity at the wall which are not a feature of two-fluid model equations. Whilst several attempts have been made to incorporate particle-wall interactions into the standard boundary conditions of two-fluid models, the boundary conditions so contrived are artificial and depend upon some a priori knowledge of the particle velocity distribution at the wall. The significant feature of the pdf approach is that it deals with the particle velocity distribution as well as that of its position so that the boundary conditions for the solution of the pdf equation are compatible with those of the natural boundary conditions. Whilst this proposal will use pdf methods to develop and solve the continuum equations of the particulate phase, the most challenging aspect is the application of pdf methods to particle-wall interactions and near wall behaviour where because of the imposed boundary conditions and the steep changes in wall turbulence, continuum equations are inapplicable. Such restrictions do not apply to the pdf approach. The strategy for calculating particle flows using a pdf approach, will be to divide the flow into near and far wall regions. In the near wall region the pdf equations will be used exclusively to obtain wall functions which incorporate the natural boundary conditions and the near wall behaviour. These wall functions will then be used to solve for the far wall behaviour where the continuum equations will be derived from the pdf method. The work proposed is both innovating and challenging from both the modelling and computational points of view: the application of the pdf method explicitly to solve for the near wall behaviour and particle wall interactions is radical and entirely new; matching the near wall solution (wall function) with the far wall flow solution will require an entirely new numerical approach, whilst the solutions of the far wall continuum equations will in themselves require the use of non-standard techniques.