**1. Introduction**

Turbulent channel particulate flows have numerous engineering applications ranging from pneumatic conveying systems to coal gasifiers, chemical reactor design and are one of the most

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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, turbulent dispersion, etc.

One of the most applicable approaches in computational fluid dynamics (CFD) is the Reynoldsaveraged Navier-stokes (RANS) equation approach, which is found in many industrial implementations and is very likely to be claimed and applied in the foreseeable future.

The procedure of analysis of the predicted fluid parameters becomes much more complicat‐ ed in case of the three-dimensional (3D) dense particulate flows, with additional inclusion of two or/and four coupling phenomena of inter-particle collisions. Within a frame of the RANS modeling, one of the challenging and advantaged theoretical approaches is the Reynolds turbulence stress model (RSTM), since *k-ε* closure model cannot describe properly the flow of complex geometry.

For the proper modeling of the particulate flows, which may include multifold processes, e.g., the particle-turbulence, or/and particle-particle, and particle-wall interactions and other relevant effects, one can apply the probability dense function (PDF) approach, which gives reasonable formalism for the closure of the governing mass and momentum equations of the particulate phase. Within the PDF approach, the closure of the governing equations of the particulate phase is based on a solution of the differential transport equations written for each particle velocity covariance, taking into account possible mechanisms of the particle-turbu‐ lence and particle-particle interactions. Such procedure is similar to the RSTM closure.

Currently, the probability dense function (PDF) approach is widely applied for the numeri‐ cal modeling of the particulate flows. The PDF models, e.g., [1, 2], contain more complete differential transport equations, which are written for various velocity correlations and consider both the turbulence augmentation and attenuation due to the presence of particles.

As opposed to the round channel flows, the rectangular or/and square channel flows, even in case of unladen flows, are considerably anisotropic with respect to the components of the turbulence energy, that is more expressed near the channel walls and corners being notable for the secondary flows. In addition, the presence of particles enhances such anisotropy. Such particulate flows are studied by RSTM, which are based on the transport equations written for all components of the Reynolds stress tensor-associated with the particulate phase.

The RSTM approach allows to completely analyze the influence of particles on axial, trans‐ verse, and spanwise components of the turbulence kinetic energy, including also possible modifications of the cross-correlation velocity moments.

A number of studies based on the RSTM approach showed its good performance and capability for simulation of the complicated flows [3], as well for the turbulent subsonic [4] and super‐ sonic flows [5] and viscoelastic flows [6].

Taulbee [7] used the RSTM approach to calculate the particle-laden shear flow by applying the direct numerical simulation (DNS). The flow Reynolds number was low (*Re* = 952). There‐ fore, the method by [7] cannot be applied to the real turbulent flows characterized by consid‐ erably higher Reynolds numbers, unlike the present numerical simulation which handles with the flow Reynolds numbers of 56,000 and 140,000.

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

One of the most applicable approaches in computational fluid dynamics (CFD) is the Reynoldsaveraged Navier-stokes (RANS) equation approach, which is found in many industrial implementations and is very likely to be claimed and applied in the foreseeable future.

The procedure of analysis of the predicted fluid parameters becomes much more complicat‐ ed in case of the three-dimensional (3D) dense particulate flows, with additional inclusion of two or/and four coupling phenomena of inter-particle collisions. Within a frame of the RANS modeling, one of the challenging and advantaged theoretical approaches is the Reynolds turbulence stress model (RSTM), since *k-ε* closure model cannot describe properly the flow of

For the proper modeling of the particulate flows, which may include multifold processes, e.g., the particle-turbulence, or/and particle-particle, and particle-wall interactions and other relevant effects, one can apply the probability dense function (PDF) approach, which gives reasonable formalism for the closure of the governing mass and momentum equations of the particulate phase. Within the PDF approach, the closure of the governing equations of the particulate phase is based on a solution of the differential transport equations written for each particle velocity covariance, taking into account possible mechanisms of the particle-turbu‐ lence and particle-particle interactions. Such procedure is similar to the RSTM closure.

Currently, the probability dense function (PDF) approach is widely applied for the numeri‐ cal modeling of the particulate flows. The PDF models, e.g., [1, 2], contain more complete differential transport equations, which are written for various velocity correlations and consider both the turbulence augmentation and attenuation due to the presence of particles. As opposed to the round channel flows, the rectangular or/and square channel flows, even in case of unladen flows, are considerably anisotropic with respect to the components of the turbulence energy, that is more expressed near the channel walls and corners being notable for the secondary flows. In addition, the presence of particles enhances such anisotropy. Such particulate flows are studied by RSTM, which are based on the transport equations written for

all components of the Reynolds stress tensor-associated with the particulate phase.

modifications of the cross-correlation velocity moments.

sonic flows [5] and viscoelastic flows [6].

The RSTM approach allows to completely analyze the influence of particles on axial, trans‐ verse, and spanwise components of the turbulence kinetic energy, including also possible

A number of studies based on the RSTM approach showed its good performance and capability for simulation of the complicated flows [3], as well for the turbulent subsonic [4] and super‐

Taulbee [7] used the RSTM approach to calculate the particle-laden shear flow by applying the direct numerical simulation (DNS). The flow Reynolds number was low (*Re* = 952). There‐ fore, the method by [7] cannot be applied to the real turbulent flows characterized by consid‐

rotation and lift forces, turbulent dispersion, etc.

340 Numerical Simulation - From Brain Imaging to Turbulent Flows

complex geometry.

The RSTM approach was also applied in the model [8]. The only difference was in the fact that the closure of equations of motion of the particulate phase was based on the Boussinesq hypothesis, where the turbulent viscosity of the particulate phase was introduced and calculated using the algebraic expressions obtained in the PDF approach [9]. However, the model [8] was inherently eclectic, since it applied the RSTM closure for a carrier flow and at the same time the Boussinesq hypothesis for the particulate phase.

Recently, Mukin and Zaichik [10] have proposed the nonlinear algebraic Reynolds stress model for the gas flow laden with small heavy particles based on the PDF approach. 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. However, the model [10] does not enable a direct solution of the differen‐ tial transport equations and it applies the *k-ε* solution.

The study of the particle dispersion that occurs in the velocity uniform shear turbulent flow assumes knowing the internal structure, general relationships, and methods of the flow generation.

Based on the analysis of results of numerous investigations of turbulent dispersion of finite inertia particles, it should be singled out three effects that are of high importance and should be considered: (i) the inertia effect implying that the dispersion of solid particles might exceed the dispersion of the fluid particles in the absence of a body force [11, 12]; (ii) the crossing trajectory effect [13] meaning that in the presence of a drift velocity a finite inertia particle will disperse less than a fluid particle; and (iii) the continuity effect [14] where the dispersion in the direction of the drift velocity exceeds the dispersion in other two directions.

In the mathematical description of the particle turbulent dispersion, there are a number of models and numerical simulations, which can be classified into the Lagrangian and Eulerian (two-fluid) approximations. They relate to the specific flow structures, such as confined flows in pipes and channels or free jets, wakes, and wall boundary flows.

Taylor [15] made the Lagrangian analysis of the dispersion of a particle in a stationary homogeneous turbulence, which showed that turbulent dispersion varies in time and derived the asymptotic expressions of dispersion for short and large times. Later on, the Lagrangian approach for the turbulent dispersion of particles was further developed in [16–21].

One of the most comprehensive numerical research of the particle dispersion in the uniform shear flow, based on the Lagrangian approach, was carried out in [22]. Ahmed and Elghoba‐ shi [22] applied DNS to the investigation of the particle inertia effect, effect of direction of gravitation as well as magnitude of a shear number. In particular, Ahmed and Elghobashi [22] have revealed that gravity reduces the particle turbulent dispersion and diffusivity in all directions due to the drift velocity effects.

On the contrary, there are very less studies on the particle turbulent dispersion, which are based on the Eulerian approach. The most significant are the numerical investigations [9, 23], where the statistical PDF models were applied to the particle behavior in the turbulent flows.

The 3D RSTM approach presented here applies the closure of equations of motion of the particulate phase, which is carried out similarly to the closure of the carrier flow, i.e., the equations are written for the normal and shear components of the Reynolds stress. The Reynolds stress equations are derived from the PDF model [9] and presented in a general case. The advantage of the given model is in use of the same closure for both the carrier flow and particulate phase, namely, the Reynolds differential equations.

The given 3D RSTM model has been applied for the turbulent dispersion of solid particles in a turbulent horizontal channel flow imposed to uniform shear.

The obtained numerical results have been verified and validated by comparison with the experimental data.
