Mesoscale model mass conservation

More often, three-dimensional gridded meteorological forecast data provided by mesoscale meteorological models such as MM5 will be available. The resolution of the grid will typically be on the order of 1 km, which is too large to capture microscale surface features. Mesoscale model results can be coupled with fine-scale models by enforcing conservation of mass to provide enhanced fine-scale detail. 

In the formulation of a mesoscale model, the number-density function (NDF) plays a key role. For this reason, we discuss the properties of the NDF in some detail in Chapter 2. In words, the NDF is the number of particles per unit volume with a given set of values for the mesoscale variables. Since at any time instant a microscale particle will have a unique set of microscale variables, the NDF is also referred to as the one-particle NDF. In general, the one-particle NDF is nonzero only for realizable values of the mesoscale variables. In other words, the realizable mesoscale values are the ones observed in the ensemble of all particles appearing in the microscale simulation. In contrast, sets of mesoscale values that are never observed in the microscale simulations are non-realizable. Realizability constraints may occur for a variety of reasons, e.g. due to conservation of mass, momentum, energy, etc., and are intrinsic properties of the microscale model. It is also important to note that the mesoscale values are usually strongly correlated. By this we mean that the NDF for any two mesoscale variables cannot be reconstructed from knowledge of the separate NDFs for each variable. Thus, by construction, the one-particle NDF contains all of the underlying correlations between the mesoscale variables for only one particle. 

As can be seen from Eq. (5.100), the virtual-mass force reduces the drag and lift forces by a factor of 1 /y. The buoyancy force is not modified because we have chosen to define it in terms of the effective volume Vpy. We remind the reader that the mesoscale acceleration model for the fluid seen by the particle A j must be consistent with the mesoscale model for the particle phase A p in order to ensure that the overall system conserves momentum at the mesoscale. (See Section 4.3.8 for more details.) As discussed near Eq. (5.14) on page 144, this is accomplished in the single-particle model by constraining the model for Apf given the model for Afp (which is derived from the force terms introduced in this section). Thus, as in Eqs. (5.98) and (5.99), it is not necessary to derive separate models for the momentum-transfer terms appearing in Apf. 

In terms of the fluid mass seen by the particle f, conservation of mass at the mesoscale leads to the following mesoscale models in the limit of zero particle Stokes number (i.e. u = U = Uf) ... 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

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. 

By resolving the structures and dominant mechanisms, it is possible to establish a stability condition reflecting the compromise between different dominant mechanisms for multiphase reaction systems, and the stability condition supplies a mesoscale constraint in addition to mass and momentum conservation equations. The calculation of EMMS model... 

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