Fluid-particle flow mesoscale model

E. Ttlzel, Particle-based mesoscale modeling of flow and transport in complex fluids, Ph.D, thesis. University of Minnesota (2006). 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

This formulation is particularly convenient when Euler-Lagrange simulations are used to approximate the disperse multiphase flow in terms of a fimte sample of particles. As discussed in Sections 5.2 and 5.3, although some of the mesoscale variables are intensive (i.e. independent of the particle mass), it is usually best to start with a conserved extensive variable (e.g. particle mass or particle momentum) when deriving the single-particle models. For example, in Chapter 4 we found that must have at least one component, corresponding to the fluid mass seen by a particle, in order to describe cases in which the disperse-phase volume fraction is not constant. 

As shown in Figure 26.1, the wide gap opens up between the particle and continuum paradigms. This gap cannot be spanned using statistical mechanical methods only. The existing theoretical models to be applied in the mesoscale are based on heuristics obtained via downscaling of macroscopic models and upscaling particle approach. Simphfied theoretical models of complex fluid flows, e.g., flows in porous media, non-Newtonian fluid dynamics, thin film behavior, flows in presence of chemical reactions, and hydrodynamic instabilities formation, involve not only vah-dation but should be supported by more accurate computational models as well. However, until now, there has not been any precisely defined computational model, which operates in the mesoscale, in the range from 10 A to tens of microns. 

---