Favre averages

Note that we have also introduced the Reynolds-average, phase-average density (p). Applying the Favre average to Eq. (157) yields a closed expression for the mass balance as follows ... 

Thus, for statistically stationary flow, the Favre-averaged velocity field satisfies V ((p)(U = 0. 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

The system of equations for gas phase was obtained by Favre averaging the system of multicomponent multiphase medium. The modified k-e model is used to describe the behavior of the gas phase. The generalization of this model will... 

Using the Favre averaging procedure [2, 5] one obtains the following set of equations for the gas phase in a multiphase flow [6] (the averaging bars are removed for simplicity) ... 

The desirability of working with Favre averages is a topic of current controversy. The controversy cannot be resolved on the basis of ease of measurement because some instruments provide v most directly, while others provide v. It is unclear whether the successful constant-density modeling of... 

By definition, P(v) possesses the necessary nonnegative and normalization properties of probability-density functions. It is especially useful in connection with the Favre-averaged formulation of the conservation equations, since corresponding averages are obtained from P(v) by the usual rules for averaging. Thus, for the v and v" defined above equation (2),... 

To illustrate how an analysis may be completed when a two-parameter representation of P(Z) is adopted, let us assume that Z obeys equation (3-71) and that p and D are unique functions of Z. Let us consider statistically stationary flows and work with Favre averages, seeking equations for the... 

For flows in which density fluctuations are negligible, the formulations become identical. Favre-averaged quantities are not easily comparable with experimentally measured quantities, which are normally non-weighted time averages. For most chemical reactor engineering applications (except maybe combustion processes), classical Reynolds averaging is suitable. 

The weighted mean quantities are defined in agreement with the standard single phase Favre averaging procedure [112, 48, 67]. That is, we define ... 

The generalized instantaneous quantity V fc is decomposed into a weighted mean component and a fluctuation component in analogy to the Favre averaging procedure for compressible flows [75, 131] ... 

Based on the Favre-averaged continuity equations (3.431), the mixture continuity equation is obtained by taking the sum over all phases ... 

At the interface the total mass is balanced in accordance with the jump mass condition (3.434), thus the RHS of (3.436) must vanish. Hence, by use of the Favre averaged form of (3.418) and (3.421) we obtain the continuity equation for the mixture (3.424). 

For systems with constant densities and no phase change, the Favre averaged continuity equation (3.431) reduces to ... 

For multiphase flows perturbed by the presence of particles to obtain a turbulence like behavior the local instantaneous velocity of the continuous phase can for example be decomposed adopting the Reynolds averaging procedure (i.e., other methods including time-, volume-, ensemble-, and Favre averaging have been used as well) and expressed as Vg = Vg- - < v >g, where v(. is the fluctuating component of the continuous phase velocity. Introducing the peculiar velocity for the dispersed phase this relation can be re-arranged as ... 

For undisturbed turbulent flows the local instantaneous velocity of the continuous phase have been decomposed in various ways, not necessarily in accordance with the familiar Reynolds - and Favre averaging procedures. 

The two-phase k — e model analyzed was based on the Favre averaged transport equation for turbulent kinetic energy developed by [73, 74]. The resulting transport equation for kinetic energy is similar to the one obtained from the single phase model (5.2), supporting the semi-empirical modification introduced in that model. 

Burns AD, Prank T, Hamill I, Shi J-M (2004) The Favre Average Drag Model for Turbulent Dispersion in Eulerian Multi-Phase Flows. 5th Int Conf Multiphase Flow, Paper No. 392, CD-ROM of ICMF 04, Yokohama, Japan, May 30-June 4... 

---