## 1. Introduction

Turbulent gas-solid particles flows in channels have numerous engineering applications ranging from pneumatic conveying systems to coal gasifiers, chemical reactor design and are one of the most thoroughly investigated subject in the area of the particulate flows. These flows are very complex and influenced by various physical phenomena, such as particle-turbulence and particle-particle interactions, deposition, gravitational and viscous drag forces, particle rotation and lift forces etc.

The mutual effect of particles and a flow turbulence is the subject of numerous theoretical studies during several decades. These studies have reported about the influence of a gas turbulence on particles (one-way coupling) and/or particles on turbulence of a carrier gas flow (the two-way coupling) in case of high flow mass loading (the four-way coupling). The influence of particles on a gas turbulence, which consists in a turbulence attenuation or augmentation depending on the relation between the parameters of gas and particles.

There are different approaches and numerical models that describe the mutual effect of gas turbulence and particles.

The *k*-*ε* models, earlier elaborated for the turbulent particulate flows, e.g., [1-5], considered a turbulence attenuation only by the additional terms of the equations of the turbulence kinetic energy and its dissipation rate. The results obtained by these models were validated by the experimental data on the turbulent particulate free-surface flows [6].

Later on, the models [7, 8] considered both the turbulence augmentation and attenuation occurring in the pipe particulate flows depending on the flow mass loading and the Stokes number. Then, these models have been expanded for the free-surface flows. As opposed to the *k*-*ε* models, [7, 8] considered both the turbulence augmentation caused by the velocity slip between gas and particles and the turbulence attenuation due to the change of the turbulence macroscale occurred in the particulate flow as compared to the unladen flow. The given approach has been successfully tested for various pipe and channel particulate flows.

Currently, the probability dense function (PDF) approach is widely applied for the numerical modeling of the particulate flows. The PDF models, for example, [9-13] contain more complete differential transport equations, which are written for various velocity correlations and consider both the turbulence augmentation and attenuation due to the particles.

As opposed to the pipe flows, the rectangular and square channel flows, even in case of unladen flows, are considerably anisotropic with respect to the components of the turbulence energy, that is vividly expressed near the channel walls and corners being notable as for the secondary flows. In addition, the presence of particles aggravates such anisotropy. Such flows are studied by the Reynolds stress turbulence models (RSTM), which are based on the transport equations for all components of the Reynolds stress tensor and the turbulence dissipation rate. RSTM approach allows to completely analyze the influence of particles on longitudinal, radial and azimuthal components of the turbulence kinetic energy, including also possible modifications of the cross-correlation velocity moments.

A few studies based on the RSTM approach showed its good performance and capability for simulation of the complicated flows, e.g., [14], as well for the turbulent particulate flows, for example, see [15]. Recently, the nonlinear algebraic Reynolds stress model based on the PDF approach has been proposed in [16] for the gas flow laden with small heavy particles. The original equations written for each component of Reynolds stress were reduced to their general form in terms of the turbulence energy and its dissipation rate with additional effect of the particulate phase. Eventually, the model [16] operated with the *k*-*ε* solution and did not allow to analyze the particles effect on each component of the Reynolds stress.

The 3D RSTM model, being presented in this chapter, is intended to apply for a simulation of the downward turbulent particulate flow in channel of the square cross-section (the aspect ratio of 1:6) with rough walls.

In order to approve and validate the developed model, the separate investigations have been carried out. The first study was the simulation of the downward unladen gas flow in channel of the rectangular cross-section with the smooth and rough walls. The second study relates to the downward grid-generated turbulent particulate flow in the same channel with the smooth walls.

The further stage of this study will be the development of the present model for implementation to the particulate channel flow with the rough walls and the initial level of turbulence.

## 2. Governing equations and numerical method

The present 3D RSTM model is based on the two-way coupling *k*-*L* model [8] and applies the 3D RANS equations and the RSTM closure momentum equations.

The sketch of the computational flow domain is shown in Figure 1 for the case of the downward grid-generated turbulent particulate flow in the channel of square cross-section. Here

### 2.1. Governing equations for the Reynolds stress turbulence model

The numerical simulation of the stationary incompressible 3D turbulent particulate flow in the square cross-section channel was performed by the 3D RANS model with applying of the 3D Reynolds stress turbulence model for the closure of the governing equations of gas, while the particulate phase was modeled in a frame of the 3D Euler approach with the equations closed by the two-way coupling model [8] and the eddy-viscosity concept.

The particles were brought into the developed isotropic turbulent flow set-up in channel domain, which has been preliminary computed to obtain the flow velocity field. The system of the momentum and closure equations of the gas phase are identical for the unladen while the particle-laden flows under impact of the viscous drag force. Therefore, here is only presented the system of equations of the gas phase written for the case of the particle-laden flow in the Cartesian coordinates.

3D governing equations for the stationary gas phase of the laden flow are written together with the closure equations as follows:

continuity equation:

where

*x*-component of the momentum equation:

(2) |

*y*-component of the momentum equation:

(3) |

*z*-component of the momentum equation:

(4) |

the transport equation of the *x*-normal component of the Reynolds stress:

(5) |

the transport equation of the *y*-normal component of the Reynolds stress:

(6) |

the transport equation of the *z*-normal component of the Reynolds stress:

(7) |

the transport equation of the *xy* shear stress component of the Reynolds stress:

(8) |

the transport equation of the *xz* shear stress component of the Reynolds stress:

(9) |

the transport equation of the *yz* shear stress component of the Reynolds stress:

(10) |

the transport equation of the dissipation rate of the turbulence kinetic energy:

(11) |

The given system of the transport equations (Eqs. 1 – 11) is based on the model [17] with applying of the numerical constants taken from [18]:

The additional terms of Eqs. (2 – 7) pertain to presence of particles in the flow and contain the particle mass concentration *u*, *v*, *w*).

The production terms *P* are determined according to [18] as follows:

(18) |

The diffusive or second order partial differentiation over Cartesian coordinates, i.e. the first three terms in Eqs. (5 – 11) are given, e.g. in [18]. The anisotropy terms

The relative friction coefficient

3D governing equations for the particulate phase are written as follows:

the particle mass conservation equation:

*x*-component of the momentum equation:

(26) |

*y*-component of the momentum equation:

(27) |

*z*-component of the momentum equation:

(28) |

The closure model for the transport equations of the particulate phase was applied to the PDF model [20], where the turbulent kinetic energy of dispersed phase, the coefficients of the turbulent viscosity and turbulent diffusion of the particulate phase are determined as follows, respectively:

(29) |

where

### 2.2. Boundary conditions for the Reynolds stress turbulence model

The grid-generated turbulent flow is vertical, and it is symmetrical with respect to the vertical axis for both *y*-and *z*-directions. Therefore, the symmetry conditions are set at the flow axis, and the wall conditions are set at the wall. In case of the rough and smooth walls the flow was asymmetrical over the *y*-direction and symmetrical over the *z*-direction.

The axisymmetric conditions are written as follows:

for *z*=0:

for

The wall conditions are written as follows:

for

for

where *s* is a roughness height. The friction velocity of gas *k*-*ε* model,

For the normal and shear stresses and dissipation rate of the unladen flow calculated at the wall, the boundary conditions are set based on the “wall-function” according to [18] with the following relationships for the production and dissipation terms:

for

for

The boundary conditions for the particulate phase are set at the wall as follows:

for

for

At the exit of the channel the following boundary conditions are set:

(40) |

Additionally, the initial boundary conditions are set for three specific cases:

### 2.3. Numerical method

The control volume method was applied to solve the 3D partial differential equations written for the unladen flow (Eqs. 1 – 11) and the particulate phase (Eqs. 26 – 29), respectively, with taking into account the boundary conditions (Eqs. 30 – 40). The governing equations were solved using the implicit lower and upper (ILU) matrix decomposition method with the flux-blending differed-correction and upwind-differencing schemes [21]. This method is utilized for the calculations of the particulate turbulent flows in channels of the rectangular and square cross-sections. The calculations were performed in the dimensional form for all the flow conditions. The number of the control volumes was 1120000.

## 3. Numerical results

The validation of the present model took place in two stages.

In case of the unladen flow, the model was validated by comparison of the kinetic (normal) components of stresses with the experimental data [22] obtained for the specially constructed horizontal turbulent gas flow in the channel of rectangular cross-section (the aspect ratio of 1:6) of 54 mm width with the smooth and rough walls for the flow Reynolds number *Re*=56000 and the roughness height of 3.18 mm.

Figure 2 shows the distributions of the longitudinal component of the averaged velocity of gas *y*/*h*=0 corresponds to the rough wall and *y*/*h*=1 corresponds to the smooth wall. The subscript “0” denotes the unladen flow conditions.

One can see that in case of the smooth channel walls, the mean flow velocity and the components of the turbulence kinetic energy demonstrate the representative symmetrical turbulent distributions over the cross-section of the rectangular channel. The transfer to the rough walls results in transformation of the given distributions. The maximum of the distribution of the time-averaged flow velocity moves towards the smooth wall. The similar change relates to the distributions of each component of the turbulence kinetic energy. These numerical results demonstrates the satisfactory agreement with the experimental data [22].

The next step of the study was the extension of the present model to the gas-solid particles grid-generated turbulent downward vertical channel flow. The experimental data [23] obtained for the channel flow of 200 mm square cross-section loaded with 700-*μ*m glass beads of the physical density 2500 *M*=4.8 and 10 mm were used for generating of the flow initial turbulence length scale.

The validity criterion was based on the satisfactory agreement of the axial turbulence decay curves occurring behind different grids in the unladen and particle-laden flows obtained by the given RSTM model and by the experiments [23]. Figure 4 demonstrates such agreement for the grid *M*=4.8 mm.

Figure 5 shows the decay curves calculated by the present RSTM model for the grids *M*=4.8 and 10 mm. As follows from Figs. 4 and 5, the pronounced turbulence enhancement by particles is observed for both grids. The character of the turbulence attenuation occurring along the flow axis agrees with the behavior of the decay curves in the grid-generated turbulent flows described in [24].

Figures 6 – 11 show the cross-section modifications of three components of the Reynolds stress, *μ*m glass beads, calculated by the presented RSTM model at two locations of the initial period of the grid-generated turbulence decay

One can see that the turbulence enhancement occupies over 75% of the half-width of the channel, that takes place at the initial period of the turbulence decay of the particle-laden flow as compared to the unladen flow. Along with, the distributions of

The distributions of modification of

The certain increase of

The analysis of Figure 13 shows that the increase of the grid mesh size results in the weaker contribution of particles to the turbulence enhancement and dissipation of the kinetic energy taking place over the cross-section for the initial period of the turbulence decay. This can be explained by the higher rate of the particles involvement into the turbulent motion due to the longer residence time that comes from the larger size of the eddies.

## 4. Conclusions

The RSTM model has been elaborated for the horizontal and vertical turbulent particulate flows in the channels of rectangular and square cross-sections with the smooth and rough walls.

The present RSTM model has been validated for the unladen channel gas flow with the rough wall. It satisfactorily described the experimental data on the averaged gas axial velocity and three components of the turbulence energy.

Further, the present model was applied to simulate the vertical grid-generated turbulent particulate channel flow. It considered both the enhancement and attenuation of turbulence by means of the additional terms of the transport equations of the normal Reynolds stress components. The model allowed to carry out the calculations covering the long distance of the channel length without using algebraic assumptions for various components of the Reynolds stress. The numerical results showed the effects of the particles and the mesh size of the turbulence generating grids on the turbulence modification that had been observed in experiments. It was obtained that the character of modification of all three normal components of the Reynolds stress taken place at the initial period of the turbulence decay are uniform almost all over the channel cross-sections. The increase of the grid mesh size slows down the rate of the turbulence enhancement which is caused by particles.