The closure problem

Generally, the closure problem reflects the idea of a spatially periodic porous media, whereby the entire structure can be described by small portions (averaging volumes) with well-defined geometry. Two limitations of the method are therefore related to how well the overall media can be represented by spatially periodic subunits and the degree of difficulty in solving the closure problem. Not all media can be described as spatially periodic [6,341 ]. In addition, the solution of the closure problem in a complex domain may not be any easier than solving the original set of partial differential equations for the entire system. 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

It is important to note that the closure problem for the species continuity equation requires solutions for the deviations of the potential, i.e the /and g fields, anfi knowledge of the average potential ( ). This result is very similar to that found by the area averaging method in Sauer et al. [345], Utilizing the closure expressions the average species continuity equation becomes... 

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

For simplicity, we will use the fictitious chemical species as the subscripts whenever there is no risk of confusion. As shown above, this reaction can be rewritten in terms of two reacting scalars and thus two components for S. The closure problem, however, cannot be eliminated by any linear transformation of the scalar variables. 

Alternatively, a joint velocity, scalar, scalar-gradient PDF could be employed. However, this only moves the closure problem to a higher multi-point level. 

In order to close (Jwe can recognize that because J(0) depends only on the 0, it is possible to replace e by (e The closure problem then reduces to finding an expression for the doubly conditioned joint scalar dissipation rate matrix. For example, if the FP model is used to describe scalar mixing, then a model of the form... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-empirical models to express the Reynolds stresses in terms of time-averaged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. 

Begin by noting that C() satisfy the Neumann problem given by (5) and boundary conditions whose solution is Cj(x, y, t) = Cj(x, t). We now derive the overall macroscopic mass balance for the species. To this end we begin by deriving the closure problem for Cj. Given Cf(x,t), combine (6) with boundary conditions and neglect the advection induced by du/dt, to obtain the local Neumann problem... 

So far, we are able to construct the constitutive equations for qc, Pc, and y. For moderate solids concentrations, we can neglect the kinetic contributions in comparison to the collisional ones. Thus, we can assume P Pc and qk qc. Substituting the constitutive relations into Eqs. (5.274), (5.275), and (5.281), after neglecting the kinetic contributions, yields five equations for the five unknowns ap, Up, and Tc (or ( 2)). Hence, the closure problem is resolved. 

A more fruitful solution to the closure problem is provided by the use of probability density functions for the fluctuating components. Various shapes (spiked, square wave, gaussian distributions) have been successfully tried (3). 

Here 0 denotes the mean temperature and 6 the local temperature fluctuation. The terms ufi represent transports of internal energy by turbulent motions, and it is these terms that bring the closure problem. 

Comparison with (1.385) shows that the equation for the mean velocity is just the Navier-Stokes equation written in terms of the mean variables, but with the addition of the term involving v -v -. Thus, the equations of mean motion involve three independent unknowns Fj, p and v -v -. This is perhaps the best known version of the closure problem. Equation (1.387) is the Reynolds equation and the term v v j is the Reynolds stress. This term represents the transport of momentum due to turbulent fluctuations. 

The basic similarity hypothesis states simply that the turbulent transport processes of momentum, heat and mass are caused by the same mechanisms, hence the functional properties of the transfer coefficients are simiiar. The different transport coefficients can thus be related through certain dimensionless groups. The closure problem is thus shifted and henceforth consist in formulating sufficient parameterizations for the turbulent Prandtl Pr )- and Schmidt (Sct) numbers. 

When the population balance is written in terms of one internal coordinate (e.g., particle diameter or particle volume), the closure problem mentioned above for the moment equation has been successfully relaxed for solid particle systems by the use of a quadrature approximation. 

As described in earlier chapters, for univariate QMOM the closure problem appears in different parts of the transport equations for the moments of the NDF. However, it can often be reduced to the following integral ... 

In univariate QMOM, the closure problem can be overcome by any numerical scheme capable of calculating the integral in Eq. (3.1). A very simple way to do this is to resort to an interpolation formula ... 

The theory of Gaussian quadrature applies only to univariate distributions. However, in practical cases, the study of distributions with multiple internal coordinates is often necessary. In these cases the closure problem generally assumes the following form ... 

The MOM was introduced for particulate systems by Hulburt Katz (1964). In their pioneering work these authors showed how it is possible to solve the PBF in terms of the moments of the NDF and to derive the corresponding transport equations. A similar approach can be used for the solution of the KF, and a detailed discussion on the derivation of the moment-transport equations can be found in the works of Struchtrup (2005) and Truesdell Muncaster (1980). The main issue with this technique is in the closure problem, namely the impossibility of writing transport equations for the lower-order moments of the NDF involving only the lower-order moments. Since the work of Hulburt Katz (1964) much progress has been made (Frenklach, 2002 Frenklach Harris, 1987 Kazakov Frenklach, 1998), and different numerical closures have been proposed (Alexiadis et al, 2004 Kostoglou Karabelas, 2004 Strumendo Arastoopour, 2008). The basis... 

A second class of methods for overcoming the closure problem is to make a functional assumption regarding the NDF. The simplest is to assume that the NDF is composed of a delta function centered on the mean value of ffie internal coordinate (e.g. mi/mo), or, in other words, assume that the population of particles is monodisperse. On resorting to this approach the missing moments can be readily calculated (e.g. mk = as illustrated... 

Quadrature-based moment methods (QBMM) constitute a particular class of very successful moment methods that overcome the closure problem by using quadrature approximations. A general (multivariate) quadrature of N nodes for M internal coordinates requires knowledge of N weights, w , and N nodes (or abscissas), y M,a),... 

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