(212a) A Favre-Averaged Turbulence Model for Gas-Particle Flows

Computational fluid dynamics (CFD) simulations are routinely used to analyze a wide range of industrial and natural processes, mostly for single phase flows. In spite of their many well known shortcomings, RANS equations are used for the majority of these single phase CFD calculations. These are usually based on a k-ε turbulence model, although many other types of turbulence models have been developed. However, multiphase CFD calculations are very costly since they are invariably transient; no adequate turbulence model has been developed for the gas-particle two-fluid model which would allow a direct calculation of the time-steady behavior of the flow. A long transient calculation must be performed from which the time-steady behavior is extracted by averaging.

The two-fluid equations are themselves derived by an averaging procedure which introduces several constitutive relations, the most important being the phase-interaction (drag) term. This presents an additional complication for a multiphase RANS theory since many different forms of these relations have been presented in the literature. For the purposes of this initial development of a RANS model for gas-solid flow, the full equation set of MFIX (Syamlal, et al., 1993) will be truncated to describe isothermal, non-reactive flow. Further, the granular pseudo-temperature equation will not be included; the granular stress will be described by a fixed viscosity as suggested by Miller and Gidaspow (1992), appropriate for dense, vertical gas-solid flow in a pipe.

For the case of multiphase flow, a RANS theory has not been extensively developed. Because of the many non-linear terms in the two-fluid theory equation set, a formal RANS theory presented in an early paper of Elghobashi and Abou-Arab (1983) contained 38 “closure” terms and the equations for multiphase turbulence kinetic energy (k equations) contained 64 such terms. In order to reduce the required number of closure relationships, Besnard and Harlow (1988) used an extension of Favre averaging which was introduced to address this same issue for compressible single phase flows (Favre, 1965; 1992). This approach has been followed for the description of bubbly flows, mainly motivated by nuclear reactor safety issues (Lahey, 2005; Krepper et al., 2005, 2009). However, an equivalent development has not occurred to describe gas-solid flows. In this paper, these ideas are developed for the two-fluid equation set of gas solid flow, as used in the open-source software MFIX, developed at the National Energy Technology Laboratory (https://mfix.netl.doe.gov). The Favre-like average of the phasic velocities are introduced and useful identities are derived from these definitions. The relationship between the Favre-averaged velocity and the Reynolds-average is discussed. The averaged continuity and momentum equations are then derived for each phase. The continuity equations have the identical form as their transient counterparts. However, the momentum equations contain many (but much less than 38) terms that require closure, including a specific turbulent Reynolds stress for each phase. Specific turbulent kinetic energies can be introduced as the contraction of the Reynolds stresses. As in single phase turbulence theory, transport equations have been derived for these turbulent stresses and turbulent kinetic energies, which include dissipative terms. It is shown that the energy of the flow can be partitioned into two quadratic terms, one involving the square of the Favre-averaged velocity and the other being the turbulent kinetic energy. Thus the energy of the flow can be partitioned between a term involving the mean flow and the fluctuations associated with volume fractions and the turbulent kinetic energy which includes the energy of the remaining fluctuations. It is clear from these equations that the turbulent energy cascade is much more complicated than single phase flow since fluctuation energy can be exchanged between phases. Most likely, energy introduced in the fluid (carrier) phase is transferred to the granular phase and dissipated there. (A very important dissipative channel, inelastic particle collisions, is not open in this initial theory since the granular temperature equation is not included.) Some suggestions are introduced for calculating the closure terms, although these are very preliminary.  

 References:

 Besnard, D.C., and F .H. Harlow, “Turbulence in multiphase flow,” Int. J. Multiphase Flow 14, 679-699, 1988.

Elghobashi, S.E., and T.W. Abou-Arab, “A two equation turbulence model for two-phase flows,” Phys. Fluids 26, 931-938, 1983.

Favre, A., “Équations des gaz turbulents compressibles I.- Formes générales,” Journal de Mécanique 4, 361-390, 1965.

Favre, A., “Formulation of the statistical equations of turbulent flows with variable density,” in Studies in Turbulence, T.B. Gatski, S. Sarkar and C.O. Speziale, eds., Springer-Verlag, 324-341, 1992.

Krepper, E., D. Lucas and H.-M. Prasser, “On the modelling of bubbly flow in vertical pipes,” Nucl. Eng. Des. 235, 597–611, 2005.

Krepper, E., M. Beyer, T. Frank, D. Lucas and H.-M. Prasser, “CFD modelling of polydispersed bubbly two-phase flow around an obstacle,” Nucl. Eng. Des. 239, 2372–2381, 2009.

Lahey Jr., R.T., “The simulation of multidimensional multiphase flows,” Nucl. Eng. Des. 235, 1043-1060, 2005

Miller, A., and D. Gidaspow, “Dense, vertical gas-solid flow in a pipe,” AIChE J. 38, 1801-1815, 1992.

Syamlal M., W. Rogers and T.J O’Brien, “MFIX documentation: theory guide,” Technical Note, DOE/METC-94/1004, NTIS/DE94000087, Springfield, VA: National Technical Information Service, 1993.