**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.

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 *u* and

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

http://dx.doi.org/10.5772/57047

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are the longitudinal components of velocities of gas and particles, respectively.

**Figure 1.** Downward channel grid-generated turbulent particulate flow.

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

by the two-way coupling model [8] and the eddy-viscosity concept.

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

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

*us*

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 implementa‐ tion to the particulate channel flow with the rough walls and the initial level of turbulence.
