## Communications in Mathematical Sciences

- Commun. Math. Sci.
- Volume 5, Issue 2 (2007), 331-353.

### A hierarchy of models for turbulent dispersed two-phase flows derived from a kinetic equation for the joint particle-gas pdf

Komla Domelevo and Philippe Villedieu

#### Abstract

This paper deals with the statistical modeling of turbulent two-phase flows consisting of particles or droplets immersed in a gas. The problem of gaseous turbulence alone being very complex, we concentrate here on the simpler case of an a priori given forced isotropic homogeneous turbulence acting on the particles, whose mean square velocity and integral Lagrangian time-scale are given constants. Our main objective is to derive a hierarchy of reduced models from the joint particle-gas pdf (probability density function). The latter equation may therefore be regarded as a
master equation for our problem. The reduced models describe the dispersion of a cloud of particles observed at di®erent time scales compared to the dynamic response time of the particles and the
characteristic time scale of the turbulence along their trajectories. These derivations rely on very classical Chapman-Enskog expansions. We recover in particular the result of Tchen [C. M. Tchen, */i>"Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid*, PhD thesis, Delft, The Hague, Martinus Nijhoff, 1947] stating that the diffusion rate is the same for small or large particles in homogeneous turbulence, under the assumption that the lagrangian statistical properties along their paths are the same. Moreover, our approach allows us to prove that the long-time limit of the joint particle-gas distribution function is a bi-maxwellian distribution, whatever the size of the particles. This is consistent with some usual assumptions made in the literature for the derivation of particle collision models [J. Laviéville, E. Deutsch and O. Simonin, *Large eddy simulation of interactions between colliding particles and a homogeneous isotropic turbulence field*, Gas-Solid Flows, ASME, 228, 347-358, 1995], [Leonid I. Zaichik, Olivier Simonin and Vladimir M. Alipchenkov, *Two statistical models for predicting collision rates of inertia particles in homogeneous isotropic turbulence*, Phys. Fluids, 15(10), 2995-3005, 2003].

#### Article information

**Source**

Commun. Math. Sci., Volume 5, Issue 2 (2007), 331-353.

**Dates**

First available in Project Euclid: 9 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.cms/1183990369

**Mathematical Reviews number (MathSciNet)**

MR2334846

**Zentralblatt MATH identifier**

1141.35318

**Subjects**

Primary: 35Q99: None of the above, but in this section 35Q99: None of the above, but in this section

**Keywords**

turbulence fluid-particles interaction hydrodynamic limits

#### Citation

Domelevo, Komla; Villedieu, Philippe. A hierarchy of models for turbulent dispersed two-phase flows derived from a kinetic equation for the joint particle-gas pdf. Commun. Math. Sci. 5 (2007), no. 2, 331--353. https://projecteuclid.org/euclid.cms/1183990369