Higher-order moment closures

Obviously, conditional moments of higher order could also be modeled. However, as with moment closures, the unclosed terms in the higher-order transport equation are more and more difficult to close. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

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. 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

Another closure approximation relies on the use of higher order moments, but this is almost a deadlock as for only two reacting components, a 13-equation model is required (3). 

This implies that higher order moments are introduced, thus the system of PDEs cannot be closed analytically. It is possible to show that similar effects will occur for the other source terms as well. This problem limits the application of the exact method of moments to the particular case where we have constant kernels only. In other cases one has to introduce approximate closures in order to eliminate the higher order moments ensuring that the transport equations for the moments of the particle size distribution can be expressed in terms of the lower order moments only (i.e., a modeling process very similar to turbulence modeling). 

Therefore, the approximate integral method is not widely used solving the population balance model for bubbly flows as the kernels involved for these systems are rather complex, thus it is very difficult to eliminate the higher order moments developing closures with sufficient accuracy. 

Inserting the same quadrature approximation to the source term integrals provides an approximate numerical type of closure avoiding the higher order moments closure problem on the cost of model accuracy [131]. This numerical approximation actually neglects the physical effects of the higher order moments. No reports applying this procedure to bubbly flows have been found so far. 

The process of finding moment-transport equations starting from the PBE can be continued to arbitrary order. We should note that in most applications the resulting moment-transport equations will not be closed. In other words, the moment-transport equation of order k will involve moments of order higher than k. We will discuss moment-closure methods in Chapters 7 and 8 in the context of quadrature-based moment methods. 

The other terms involving -]p are defined similarly. Note that, as is usually the case in moment methods, the convection term is unclosed and involves even higher-order moments. The remaining terms on the right-hand side of Eq. (4.109) are defined in a manner very similar to their counterparts in Eq. (4.85). Thus we will not discuss them in detail except to say that they will usually not appear in closed form. Only in the special case discussed in Section 4.4 where the velocity NDF is nearly Gaussian can we provide an accurate closure for Eq. (4.111) in terms of the velocity moments of up to second order. 

The moment equations (7) and (10) depend on a higher order moment, 3, which requires a closure method in order to solve the moment differential equations. The closure method used is that of Hulburt and Katz 1QJ according to which pj is expressed as... 

Hulburt and Katz [7] have developed a method to obtain estimates of the higher moments in terms of lower ones using a distribution approximation method. Most previous work has been based on the 3rd-moment closure. A general expression is constructed for moment /< , where I is the highest order of the series of Laguerre polynomials in the approximation, while a and y are parameters to be specified ... 

The outcome of this procedure is, in addition to the previously described kinetic theory of granular flow (KTGF) transport equations with the given flux and source closures, characteristic transport equations for species mass and thermal temperature. It is noted that the use of second order velocity moments or higher moments usually requires some kind of manipulation in order to obtain equations in the desired form. The derivation of the thermal temperature equation for reactive systems is certainly not trivial. The application of this theory to reactive systems is extensively discussed in the following two sections. 

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