Mesoscale variability

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

Sherr, E. B., B. F. Sherr, and T. J. Cowles. 2001. Mesoscale variability in bacterial activity in the Northeast Pacific Ocean off Oregon, USA. Aquatic Microbial Ecology 25 21—30. 

The mesoscale variability in the Black Sea is mostly related to the meandering of the RC, to the formation of anticyclonic eddies between the RC and the coast, to their transformation into the deep-sea eddies, and to the interaction of the latter with the neighboring anticyclonic and cyclonic circulation elements and with the RC. The meandering jet of the RC together with the... 

Vostokov SV, Lisitsyn BE, Konovalov BV, Soloviev DM, Gagarin VI (2002) Mesoscale variability of chlorophyll a concentration, particulate organic matter content and spectral index of light absorption by phytoplankton in the upper layer of northeastern part of the Black Sea. In Zatsepin AG, Flint MV (eds) Multi-disciplinary investigations of the northeastern part of the Black Sea. Nauka, Moscow, p 235 (in Russian)... 

As illustrated in Figure 1.2, the key modeling steps in the mesoscale approach are (1) defining the mesoscale variables and (2) deriving a model for how one particle s mesoscale variables change due to the microscale physics involving all particles. In some cases, the choice of the mesoscale variables is straightforward. For example, due to mass and... 

Define phase space of mesoscale variables needed to describe a particle (velocity, volume, etc.)... 

Model changes to one particle s mesoscale variables due to all other particles, fluid, body forces, etc. (one-particle density function)... 

In summary, the microscale description provides two important pieces of information needed for the development of mesoscale models. First, the mathematical formulation of the microscale model, which includes all of the relevant physics needed to completely describe a disperse multiphase flow, provides valuable insights into what mesoscale variables are needed and how these variables interact with each other at the mesoscale. These insights are used to formulate a mesoscale model. Second, the detailed numerical solutions from the microscale model are directly used for validation of a proposed mesoscale model. When significant deviations between the mesoscale model predictions and the microscale simulations are observed, these differences lead to a reformulation of the mesoscale model in order to improve the physical description. Note that it is important to remember that this validation step should be done by comparing exact solutions to the mesoscale model with the microscale results, not approximate solutions that result... 

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 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. 

In summary, the complexity of the correlations between the mesoscale variables describing polydisperse multiphase flows requires us to employ a mathematical formulation that is capable of exactly describing such correlations. The one-particle NDF contains all... 

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... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

The second example is the case in which the disperse-phase velocity is equal to the conditional expected disperse-phase velocity given the other mesoscale variables Vp = (Upl p, Vf, f). In this case, fluctuations in the disperse phase are slaved to the fluid-phase fluctuations and can depend on the particle internal coordinates (e.g. particle size). The NDF in this case is n(Vp, p, Vf, f) = n( p, Vf, f)(5(Vp - [Pg.131]

A popular method for closing a system of moment-transport equations is to assume a functional form for the NDF in terms of the mesoscale variables. Preferably, the parameters of the functional form can be written in closed form in terms of a few lower-order moments. It is then possible to solve only the transport equations for the lower-order moments which are needed in order to determine the parameters in the presumed NDF. The functional form of the NDF is then known, and can be used to evaluate the integrals appearing in the moment-transport equations. As an example, consider a case in which the velocity NDF is assumed to be Gaussian ... 

Following a single particle, the rates of change of the mesoscale variables can be written in a Lagrangian form ... 

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 a concrete example of the diffusion coefficients, let us consider a case in which the only mesoscale variables are U and U. Moreover, let us assume that the Lagrangian... 

In the GPBE, the masses will be replaced by the phase densities and f, which must be constant if the only mesoscale variables are U and U. ... 

---