**4. Results and discussions**

The numerical results presented below have been obtained at two locations of the flow: initial location signed "ini" and disposed at the exit of the particle source point and the location 2*x/hy*= 12.63 from the exit of the particle point source. The turbulent dispersion of 55-μm glass spherical particles was examined. The flow mass loading was about 10−6 kg dust/kg air.

**Figures 3**–**15** show the numerical data obtained by the presented model for two cases of spatial orientation of shear of the mean flow velocity: shear is along the direction of gravity (case 1), and shear is directed normally to gravity (case 2).

**Figure 3.** Transverse distributions of axial velocities of gas and particles, case 1. Here and below *u*0 is the mean flow velocity; *u*0 = 5.1 m/s.

**Figure 4.** Spanwise distributions of axial velocities of gas and particles, case 2.

Two-Fluid RANS-RSTM-PDF Model for Turbulent Particulate Flows http://dx.doi.org/10.5772/63338 355

**Figure 5.** Transverse distributions of a turbulence kinetic energy; cases 1 and 2.

**4. Results and discussions**

354 Numerical Simulation - From Brain Imaging to Turbulent Flows

and shear is directed normally to gravity (case 2).

**Figure 4.** Spanwise distributions of axial velocities of gas and particles, case 2.

velocity; *u*0 = 5.1 m/s.

The numerical results presented below have been obtained at two locations of the flow: initial

location signed "ini" and disposed at the exit of the particle source point and the location

2*x/hy*= 12.63 from the exit of the particle point source. The turbulent dispersion of 55-μm glass

**Figures 3**–**15** show the numerical data obtained by the presented model for two cases of spatial

orientation of shear of the mean flow velocity: shear is along the direction of gravity (case 1),

**Figure 3.** Transverse distributions of axial velocities of gas and particles, case 1. Here and below *u*0 is the mean flow

spherical particles was examined. The flow mass loading was about 10−6 kg dust/kg air.

**Figure 6.** Spanwise distributions of a turbulence kinetic energy, cases 1 and 2.

**Figure 7.** Transverse distributions of *xy* shear stress component of the Reynolds stress of gas and particles, case 1. Here *u* ′ *v* ¯′ =*u* ′ *v* ¯′ / *u*<sup>0</sup> 2 .

**Figure 8.** Spanwise distributions of *xz* shear stress component of the Reynolds stress of gas and particles, case 2. Here *u* ′ *w*¯′ =*u* ′ *w*¯′ / *u*<sup>0</sup> 2 .

**Figure 9.** Transverse distributions of x-normal components of the Reynolds stress of gas and particles, case 1. Here *u* ¯′2 =*u* ¯′2 / *u*<sup>0</sup> 2 and *u* ′ s ¯2 =*u* ′ s ¯2 / *u*<sup>0</sup> 2 .

**Figure 10.** Transverse distributions of particles mass concentration, case 1.

Two-Fluid RANS-RSTM-PDF Model for Turbulent Particulate Flows http://dx.doi.org/10.5772/63338 357

**Figure 11.** Spanwise distributions of axial velocities of gas and particles, case 1, location 2*x*/*hy*= 12.63.

**Figure 12.** Spanwise distribution of particles mass concentration, case 1.

**Figure 8.** Spanwise distributions of *xz* shear stress component of the Reynolds stress of gas and particles, case 2. Here

**Figure 9.** Transverse distributions of x-normal components of the Reynolds stress of gas and particles, case 1. Here

**Figure 10.** Transverse distributions of particles mass concentration, case 1.

*u* ′ *w*¯′ =*u* ′ *w*¯′ / *u*<sup>0</sup> 2 .

356 Numerical Simulation - From Brain Imaging to Turbulent Flows

*u* ¯′2 =*u* ¯′2 / *u*<sup>0</sup> 2 and *u* ′ s ¯2 =*u* ′ s ¯2 / *u*<sup>0</sup> 2 .

**Figure 13.** Transverse distribution of particles mass concentration, case 2.

**Figure 14.** Spanwise distribution of particles mass concentration, case 2.

**Figure 15.** Transverse distributions of axial velocities of gas and particles, cases 1 and 2, location 2*x*/*hy*= 12.63.

**Figure 3** shows the transverse distributions of axial velocities of gas and particles for case 1. It is evident that the linear profiles of the averaged axial velocity components of gas and particulate phase across the flow are almost preserved starting from the initial cross section till the pipe exit. Besides, they occupy almost the whole turbulent core of the flow with slight increase of the values in the turbulent core and decrease near the walls due to the effect of a viscous dissipation. The similar profiles are observed with respect of distribution of the same averaged axial velocity components for gas and particulate phase along the spanwise direction (**Figure 4**).

Since the axial velocity increases toward the bottom wall, the profiles of a turbulence kinetic energy have their higher values near the bottom wall area (**Figure 5**). However, along the spanwise direction, the profiles of the turbulence kinetic energy are symmetrical, since there is no change of the axial velocity along this direction (**Figure 6**).

The profiles of the Reynolds shear stresses of gas and particulate phase are shown in **Figures 7** and **8**. Here it is evident that there is some kind of plateau in the turbulent core. This con‐ firms that we deal with the shear flow; hence, it must be the constant value of the Reynolds shear stresses observed for cases 1 and 2, i.e., for the *xy*-plane (case 1) and *xz*-plane (case 2) Reynolds shear stress components. Here, the linear distributions of the averaged axial velocity components across the flow take place along the spanwise direction.

**Figure 9** show the transverse distributions of x-normal components of the Reynolds stress of gas and particulate phase obtained for case 1. It can be seen that unlike *<sup>u</sup>* ′ ¯2 , the maximum value of *u* ′ *s* ¯2 distribution located near the channel top wall is larger than the one near the bottom wall. This is due to the effect of particle inertia and their crosswise motion that cause different axial particle accelerations near the top and bottom walls (**Figure 3**).

**Figures 10**–**13** present the transverse and spanwise distributions of the particle mass concen‐ tration *c*/*c*0 across the flow at the initial location and the location 2*x*/*hy*= 12.63 for both the cases of spatial orientation of shear of the mean flow velocity. Here *c*0 is the value of the particle mass concentration at the initial location at the flow axis. These distributions reflect the character of the particle turbulent dispersion that occurs in the given channel shear flow. It is obvious that a) due to gravity the particles go down, and thus the mass concentration profile shifts toward the bottom wall (case 1) and b) the profiles become wider relative to their initial distribu‐ tions due to the particle turbulent dispersion (**Figure 10**).

Since in case 1 there is symmetrical distribution of parameters along the spanwise direction (**Figures 6** and **11**), the symmetrical distribution of the mass concentration along this direc‐ tion (**Figure 12**) can be observed, both at the initial and exit cross sections.

A similar situation is observed for case 2, when the linear change of the axial velocity takes place along the spanwise direction. Here the particles go down due to gravity (see **Figure 13**), and simultaneously there is no shift of the distribution of the mass concentration along the spanwise direction (**Figure 14**).

**Table 1** presents the values of the particle spatial displacement *Dy* obtained experimentally and numerically for two cases of spatial orientation of shear of the mean flow velocity. This displacement characterized quantitatively the particle turbulent dispersion. It is evident that the numerical values of displacement fit satisfactory with the experimental ones that vali‐ date the reliability of the presented model.


**Table 1.** Particle displacement.

**Figure 14.** Spanwise distribution of particles mass concentration, case 2.

358 Numerical Simulation - From Brain Imaging to Turbulent Flows

direction (**Figure 4**).

**Figure 15.** Transverse distributions of axial velocities of gas and particles, cases 1 and 2, location 2*x*/*hy*= 12.63.

**Figure 3** shows the transverse distributions of axial velocities of gas and particles for case 1. It is evident that the linear profiles of the averaged axial velocity components of gas and particulate phase across the flow are almost preserved starting from the initial cross section till the pipe exit. Besides, they occupy almost the whole turbulent core of the flow with slight increase of the values in the turbulent core and decrease near the walls due to the effect of a viscous dissipation. The similar profiles are observed with respect of distribution of the same averaged axial velocity components for gas and particulate phase along the spanwise

Since the axial velocity increases toward the bottom wall, the profiles of a turbulence kinetic energy have their higher values near the bottom wall area (**Figure 5**). However, along the spanwise direction, the profiles of the turbulence kinetic energy are symmetrical, since there

The profiles of the Reynolds shear stresses of gas and particulate phase are shown in **Figures 7** and **8**. Here it is evident that there is some kind of plateau in the turbulent core. This con‐

is no change of the axial velocity along this direction (**Figure 6**).

**Table 1** shows that the particle dispersion in case 1 is smaller than in case 2. This fact can be explained by the particle axial velocity taking place in case 2 is smaller than the one for case 1 in the same y location (**Figures 10**, **13**, and **15**).
