Meso-scale modelling number

A number of authors, however, have represented forest or urban canopy layers by porous regions of distributed force (or drag) [206, 213, 217, 318, 320, 576, 662], The advantage of such an approach is that it permits inclusion of a canopy sublayer without the use of excessive and costly grid resolution. Yamada [662] and Shaw and Schumann [576] introduced the approach in order to add vegetation to meso-scale models of complex terrain. Jeram et al. [320] used the concept in 2d calculations for inviscid flow and constant eddy diffusivity flow estimates of the up and downwind penetration of flow within simple urban areas. 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

In order to construct mesoscopic models, we again begin by partitioning the system into cells located at the nodes of a regular lattice, but now the cells are assumed to contain some small number of molecules. We cannot use a continuum description of the dynamics in a cell as we did for the reaction-diffusion equation. Instead, we describe the reactions and motions of molecules using stochastic rules that mimic the dynamics of these processes on meso-scales. The stochastic element arises because we do not take into account the detailed motions of all solvent species or the dynamics on microscopic scales. Nevertheless, because the number of molecules in a cell may be small, we must account for the fact that this number can change by random reactive events and random motions of molecules that take them into and out of a... 

The findings from the meso-scale and micro-scale DNS are used to validate the simplifications made in a RANS type engineering model. From a comparison made between RANS simulations and meso scale DNS, they conclude that the RANS model had shortcomings in accurately predicting local Nusselt (heat transfer) and Sherwood (mass transfer) numbers, but performed satisfactorily accurate in predicting global heat and mass transfer. 

A number of direct ways for linking atomistic and meso-scale melt simulations have been proposed more recently. The idea behind these direct methods is to reproduce structure or thermodynamics of the atomistic simulation on the meso-scale self-consistently. As this approach is an optimization problem, mathematical optimization techniques are applicable. One of the most robust (but not very efficient) multidimensional optimizers is the simplex optimizer, which has the advantage of not needing derivatives, which are difficult to obtain in the simulation. The simplex method was first applied to optimizing atomistic simulation models to experimental data. We can formally write any observable, like, for example, the density p, as a function of the parameters of the simulation model Bj. In Eq. [2], the density is a function of the Lennard-Jones parameters. 

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