1 Introduction

When studying turbulent gas-particle flows, mostly, either an Eulerian-Lagrangian approach, or an Eulerian-Eulerian (two-fluid) approach is used. Both of them are based on solving the Navier-Stokes equations in continuum mechanics. Alternatively, some investigators considered the particles as a randomly-moved statistical group, used the kinetic theory to derive the probability density function (PDF) transport equation for particles, and took statistical averaging to obtain the particle Reynolds stress equation in two-fluid modeling. These statistically averaged equations are close to those obtained by Reynolds averaging or mass-weighed averaging in Reynolds averaged two-fluid modeling. Still in the 1960s, Williams^[1] and Zhou^[2] independently proposed the statistical conservation equations for a particle group. However, at that time it was not related to turbulence. In the 1980s, Pope^[3] proposed the PDF transport equation for single-phase reacting flows, using the Monte-Carlo (MC) method to solve the PDF equation for turbulent combustion. In the 1990s, Derevich and Zaichik^[4], Reeks^[5], and Simonin^[6] independently derived the particle PDF equations for turbulent gas-particle flows in the particle velocity space, using these PDF equations to obtain the particle Reynolds stress equation in two-fluid modeling. Afterwards, Zhou et al.^[7] derived the joint PDF equation in the gas-particle velocity space. Unlike other investigators, Zhou et al. directly solved the PDF equations using either a finite-difference method or an MC method. The so-called DSM-PDF (DSM denotes the differential stress model) model and SOM-MC model (SOM denotes the second-order moment) were proposed. The PDF distribution for complex swirling and separating turbulent gas-particle flows was obtained, and was used to directly obtain the particle Reynolds stress and particle turbulent kinetic energy. The PDF modeling-SOM-MC method was also used to simulate turbulent evaporating gas-droplet flows. This paper gives a review of these studies.

2 PDF equations for turbulent gas-particle flows

Let us consider the statistical conservation equations for a particle group in laminar two-phase flows. In a most general case for a dispersed system, like a particle/bubble/droplet group, each particle has its own size, velocity, temperature, and material density, and particles of the same size may have different velocities, temperatures, and material densities. The PDF for the particle number density in the range of

can be defined as

Based on the Liouville theorem in statistical mechanics for a collision-less particle group, the volumetric changing rate in the phase space should be zero. Therefore, the PDF conservation equation in the phase space is

(1)

This equation is not related to turbulence, and in general it is difficult to be solved in its original form due to too many variables. Subsequently, in order to solve this equation, it is simplified for the cases of equal particle velocity or equal particle temperature or equal particle density.

Now, let us consider turbulent gas-particle flows. For simplicity, using the capital letters to denote the instantaneous values, the lower-case letters to denote the fluctuation values, and the symbol 〈*〉 to denote the statistically averaged values, the PDF for an instantaneous variable at the time instant t and the location x_j can be defined by

where p_s is the joint PDF in the gas-particle velocity space, and V_i and V_pi are the gas and particle velocity coordinates, respectively. Based on the property of the δ function, the one possible realization of p_s can be defined by

Its statistically averaged value in the phase space is

where the symbol 〈*〉 expresses the statistically averaged value after tests of infinite time, namely, the mathematical expectation of p_s'. Similarly, the PDF for a fluctuation variable in the turbulent gas-particle flow field can be defined by

where v_i and v_pi are the gas and particle fluctuation velocity coordinates, respectively, and also we have

It can be seen that, from the property of the PDF for any physical variable Q=Q(V_i, V_pi) or q=q(v_i, v_pi), the statistically averaged value in the geometrical space should be

For dilute gas-particle flows, neglecting the forces acting on the particles other than the drag force and using the instantaneous two-phase flow equations, the transport equations for p_s and p_f can be obtained as

(2)

(3)

where and express the gas and particle fluctuation drags, respectively, U_j and U_pj are gas and particle instantaneous velocities, respectively, U_j=〈U_j〉+u_j, U_pj=〈U_pj〉+u_pj, and u_j and u_pj are gas and particle fluctuation velocities, respectively.

3 SOM two-phase turbulence model

The SOM two-phase turbulence model or the two-phase Reynolds stress model for two-fluid modeling was developed based on the Navier-Stokes equations and Reynolds time averaging by Zhou and Chen^[10]. It is expressed by the following equations:

(4)

where D_ij, P_ij, Π_ij, and ε_ij are

The source term expressing the effect of particles on the gas Reynolds stress is

The closed transport equation of the dissipation rate of the gas turbulent kinetic energy is

(5)

where

The closed particle Reynolds stress equation is

(6)

where

where N_p and n_p denote the particle number density and its fluctuation value, respectively. The closed transport equations for and can be found in Ref. [10]. For example, the equations of two-phase velocity correlation and particle turbulent kinetic energy are

(7)

(8)

where

Equations (4)-(8) are called the unified SOM (USOM) or SOM two-phase turbulence model.

4 DSM-PDF two-phase turbulence model

Our approach is to directly solve the particle PDF equation without using the particle Reynolds stress equation. Two methods are used to solve the particle PDF equation. One is using the finite-difference method for two-dimensional (2D) flows, and the other one is using the MC method for three-dimensional (3D) flows. For the first method, the gas-phase k-ε model or the Reynolds-stress equation model, i.e., the DSM, is still used, and the particle Reynolds stresses are obtained by the integration over the PDF. Therefore, the two-phase turbulence models are called k-ε-PDF and DSM-PDF models. While for the second method, the PDF values and the integration over the PDF are not required, and the particle Reynolds stresses are directly obtained from the MC simulation, which is a Lagrangian method of particle motion in the phase space. Since the modeling of particle phase is more important in the simulation of turbulent gas-particle flows, and solving PDF equations is more expensive, the PDF transport equation model, i. e., the PDF, is applied only to the particle phase, and the k-ε model or Reynolds stress equation model can still be used for the gas phase. Integrate Eq. (3) in the gas fluctuating velocity space, and use the gradient modeling for the term of fluctuating drag force. The closed form of the particle-phase PDF transport equation of p_fp can be obtained as

(9)

If p_fp is obtained by solving Eq. (9), then the particle Reynolds stress and turbulent kinetic energy can be obtained directly by the integration over the PDF without solving their transport equations, that is,

(10)

(11)

Equations (9)-(11) constitute the PDF model of the particle phase. The DSM-PDF two-phase turbulence model, that is a combination of the DSM model for the gas-phase turbulence in two-phase flows and the PDF model for the particle phase, proposed by Zhou and Li^[9], is used to simulate swirling gas-particle flows. Figures 1 and 2 show the predicted particle tangential time-averaged and root mean square (RMS) fluctuation velocities with the DSM-PDF model, respectively, and their comparison with experimental results and the k-ε-k_p modeling results. It is seen that, in the case of weakly swirling flows, the DSM-PDF model gives the results close to those obtained using the k-ε-k_p model, and both of them are in agreement with the experimental results.

5 SOM-MC modeling of swirling gas-particle flows

An SOM-MC modeling proposed by Liu et al.^[11] is a combination of the gas-phase SOM (the Reynolds stress equation) model with an MC method to solve the particle Lagrangian PDF equation. The stochastic equations of particle motion and the gas-eddy motion seen by particles in the Lagrangian coordinate can be written as

(12)

(13)

(14)

The Langevin equation of gas-eddy motion seen by particles is

(15)

where the first three terms on the right-hand side of Eq. (15) reflect the effect of time-averaged gas velocity field on the stochastic motion of gas eddies, the fourth term is the additional force caused by the difference between the trajectories of particles and gas eddies, and the fifth and last terms express the effect of viscosity, fluctuating pressure, and particle motion, while the coefficient G_{gp, ij} needs further consideration. G_{gp, ij} can be modeled as

where τ_Lp is the Lagrangian integral time scale of gas seen by particles, which is different from τ_L, the Lagrangian integral time scale of gas itself. By considering the crossing-trajectory effect and continuity effect, G_{gp, ij} can be written as follows:

where τ_{Lp, //} and τ_{Lp, ⊥} are the Lagrangian integral time scales of gas seen by particles parallel and perpendicular to the trajectories of particles, respectively. According to the experiments of particle dispersion in the homogenous turbulence, we have

where C_β = 0.45, and τ_L is the Lagrangian integral time scale of gas itself and is defined as

For the gas-phase SOM model in the SOM-MC two-phase turbulence model, the gas-phase Reynolds stress equation in two-phase flows for the USOM model is used. The SOM-MC model is used to simulate swirling gas-particle flows. Figures 3 and 4 show the predicted particle tangential time-averaged and RMS fluctuation velocities with the SOM-MC model, respectively, and their comparison with the experimental results and the USOM modeling results. Obviously, there is only a slight difference between these two modeling results. Hence, the PDF equation model validates the USOM model.

6 PDF modeling of evaporating gas-droplet flows

The joint PDF equation for combusting turbulent gas-particle/droplet flows is derived and obtained as^[12]

(16)

The notations in Eq. (16) can be found in Ref. [12]. The evaporating gas-droplet flows are solved using the SOM-MC simulation. The predicted droplet velocity, droplet size and droplet mass flux and their comparison with experimental results are given in Figs. 5, 6, and 7, respectively. It is seen that the simulation results are in good agreement with the experimental results.

7 Conclusions

(ⅰ) The joint PDF transport equations in gas-particle spaces are proposed.

(ⅱ) The closed particle PDF equation is directly solved using either the finite-difference method or the MC method.

(ⅲ) The PDF modeling of particle phase, including the DSM-PDF and SOM-MC models, validates the USOM two-phase turbulence model in Reynolds-averaged modeling of turbulent gas-particle flows.

(ⅳ) Turbulent evaporating droplet flows can be well simulated using the PDF modeling.