The Ensemble Averaging Procedure

In this section we examine the ensemble averaging method in the framework of the generic single averaging procedure proposed by Drew [54] and further elaborated in [55, 137, 56, 58]. The paper by Enwald et al [67] summarizes this approach nicely in the setting of fluidized bed reactor simulations. 

Other statistical averaging methods exist as well, some of them have been assessed by [27, 104, 28, 3, 126, 247, 248, 249, 169, 142]. These reports might be recommended for complementary studies. 

As distinct from the time and volume averaging procedures, the application of the ensemble averaging operator is not restricted by any space- and time scales. The ensemble averaging is thus considered the fundamental method [58, 59]. 

Consider a local instantaneous scalar, vector or tensor valued function defined by and let ( )e(r, t) denote the corresponding averaged 

In order to avoid mathematical difficulties when applying the averaging operators to the generic equation we introduce the phase indicator function Xfc(r,t p) in any realization. This Heaviside function picks the phase k while ignoring the other phases and the interface. The characteristic function is defined by  

For some turbuient flows, the boundary conditions and initial conditions cannot be controlled suiBciently to allow repeatable experiments. In this case, although turbulent flows are not really deterministic, a useful conceptualization of the ensemble average assumes that the flow is deterministic but that randomness may arise through the uncertainty in the initial tmd boundary conditions [62]. 

Another possible conceptualization of the ensemble average imagines that the process is affected by small random forces through the motion. Particulate flows can then be described by distributions of positions, velocities and sizes adopting the basic principles of kinetic theory [91, 182, 262-264]. This alternative ensemble averaging approach is examined in relation to granular flows in Chap. 4. 

The Xjt is considered a generalized function which is especially useful in connection with differentiation of discontinuous functions within an integral. The ensemble average of is then defined by  

From the definition of a probability density function we note  

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