Mesoscale model closure

Lenz C-J, Muller F, Schliinzen KH (2000) The sensitivity of mesoscale chemistry transport model results to boundary values. Environ Monit Assess 65 287-298 L6pez SD, Liipkes C, Schliinzen KH (2005) The effects of different k-e-closures on the results of a micro-scale model for the flow in the obstacle layer. Meteorol Z 14 839-848 Muller F, Schliinzen KH, Schatzmann M (2000) Test of numerical solvers for chemical reaction mechanisms in 3D air quality models. Environ Model Softw 15 639-646 Schliinzen KH (1990) Numerical studies on the inland penetration of sea breeze fronts at a coastline with tidally flooded mudflats. Beitr Phys Atmos 63 243-256 Schliinzen KH, Katzfey JJ (2003) Relevance of subgrid-scale land-use effects for mesoscale models. Tellus 55A 232-246... 

Mesoscale model Incorporates more microscale physics in closures... 

As discussed in Chapter 2, the one-particle NDF does not usually provide a complete description of the microscale system. For example, a microscale system containing N particles would be completely described by an A-particle NDF. This is because the mesoscale variable in any one particle can, in principle, be influenced by the mesoscale variables in all N particles. Or, in other words, the N sets of mesoscale variables can be correlated with each other. For example, a system of particles interacting through binary collisions exhibits correlations between the velocities of the two particles before and after a collision. Thus, the time evolution of the one-particle NDF for velocity will involve the two-particle NDF due to the collisions. In the mesoscale modeling approach, the primary physical modeling step involves the approximation of the A-particle NDF (i.e. the exact microscale model) by a functional of the one-particle NDF. A typical example is the closure of the colli-sionterm (see Chapter 6) by approximating the two-particle NDF by the product of two one-particle NDFs. 

Chapter 1 introduces key concepts, such as flow regimes and relevant dimensionless numbers, by using two examples the PBE for fine particles and the KE for gas-particle flow. Subsequently the mesoscale modeling approach used throughout the book is explained in detail, with particular focus on the relation to microscale and macroscale models and the resulting closure problems. 

In Chapter 4 the GPBE is derived, highlighting the closures that must be introduced for the passage from the microscale to the mesoscale model. This chapter also contains an overview of the mathematical steps needed to derive the transport equations for the moments of the NDF from the GPBE. The resulting moment-closure problem is also throughly discussed. 

Now the whole set of SFM hydrodynamic equations with EMMS mesoscale modeling is estabhshed, where the drag and stress closures are based on the bimodal distribution. However, the mass exchange (or phase change) between the dilute and dense phases is still a big challenge to the kinetic theory derivation. More elaborate efforts are needed on this topic. 

In previous sections, we have shown the mesoscale modeling based on the bimodal structure. In what follows, we will show the advantage of such approaches over the ones based on local equilibrium and homogeneous closures, in particular, TFMs. The comparison starts from a simple one-dimensional force balance analysis, aiming to shed light on which kind... 

In most cases, closure of the terms in the kinetic equation will require prior knowledge of how the mesoscale variables are influenced by the underlying physics. Taking the fluid-drag term as an example, the simplest model has the form... 

In order to account for variable particle numbers, we generalize the collision term iSi to include changes in IVp due to nucleation, aggregation, and breakage. These processes will also require models in order to close Eq. (4.39). This equation can be compared with Eq. (2.16) on page 37, and it can be observed that they have the same general form. However, it is now clear that the GPBE cannot be solved until mesoscale closures are provided for the conditional phase-space velocities Afp)i, (Ap)i, (Gp)i, source term 5i. Note that we have dropped the superscript on the conditional phase-space velocities in Eq. (4.39). Formally, this implies that the definition of (for example) [Pg.113]

Here, Pc is the mixture density of the dense phase. U up i is defined by J Uf-U/), where Uf and U are mean velocities of the dilute and dense phases, respectively. This definition of mesoscale slip velocity differs a little bit from that in the cluster-based EMMS model, because the continuous phase transforms from the dilute phase to the dense phase. And their quantitative difference is l-f)PgUgc/Pc, which is normally negligible for gas-solid systems. Similarly, the closure of Fdi switches to the determination of bubble diameter. And it is well documented in literature ever since the classic work of Davidson and Harrison (1963). Compared to cluster diameter, bubble diameter arouses less disputes and hence is easier to characterize. The visual bubbles are normally irregular and in constantly dynamic transformation, which may deviate much from spherical assumption. Thus, the diameter of bubble here can also be viewed as drag-equivalent definition. 

A hindcast study has shown that the occurrence of significant seiche episodes in Rotterdam Harbor can be accurately predicted on the basis of the prediction of the occurrence of meteorological conditions conducive to the formation of mesoscale atmospheric convection cells behind a cold front. This is presently realized using output of air temperatures and moisture content from numerical atmospheric models that are used in operational weather forecasting. The outcomes of this prediction method are used by the port authorities for ship traffic control and, after further development and testing, can eventually be incorporated into the closure-management system of the storm surge barrier in Rotterdam Waterway. 

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