**3. Numerical results**

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

*y y y*

*z z z*

ll

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

ll

*u w*

*u v*

, , , 0, *<sup>s</sup> <sup>s</sup> s s s s*

; , 0; *<sup>s</sup> <sup>s</sup> s s s s*

*u v Dw*

22 2

*x x xx x x x*

**1.** the low level of the initial intensity of turbulence that usually occurs at the axis of the

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 fluxblending 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

¶¶¶ ¢¢ ¢¢ ¶¶¶ ¶ = = == = = = = ¶ ¶ ¶¶ ¶ ¶ ¶

*u v w u v w uv xx x x x x x uw vw uvw*

¶¶¶ ¶ ¶ ¶ ¶ ¢ ¢ ¢ ¢¢ == = = = = = ¶¶¶ ¶ ¶ ¶ ¶

e

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

**2.** the high level of the initial turbulence generated by two different grids:

a

*u w Dv*

a

 a¶ ¶ ¶ =- =- = = ¶¶ ¶ (38)

 a¶ ¶ ¶ =- =- = = ¶ ¶ ¶ (39)

0. *ss s*

a

(40)

for *y* =0.5*h* :

52 Computational and Numerical Simulations

for *z* =0.5*h* :

channel turbulent flow;

**2.3. Numerical method**

**a.** small grid of the mesh size *M*=4.8 mm;

**b.** large grid with mesh size of *M*=10 mm.

conditions. The number of the control volumes was 1120000.

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.** Numerical and experimental [22] distributions of the longitudinal component of the averaged velocity of gas over the channel cross-section.

Figure 2 shows the distributions of the longitudinal component of the averaged velocity of gas *u*0 over the channel cross-section for two cases: i) the smooth walls and ii) the left wall is rough and the right wall is smooth for the mean flow velocity 15.5 m/s. Figure 3 shows the distribu‐ tions of the normalized Reynolds normal stress tensor components obtained for the same conditions as Figure 2. The radial distance *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 compo‐ nents 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].

**Figure 3.** Numerical and experimental [22] distributions of the normalized Reynolds normal stress tensor components.

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 kg/m<sup>3</sup> was used for the model validation. The mean flow velocity was 9.5 m/s, the flow mass loading was 0.14 kg dust/kg air. The grids of the square mesh size *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, *Δu*, *Δv*, *Δw*, caused by 700-*μ*m glass beads, calculated by the presented RSTM model at two locations of the initial period of the grid-generated turbulence decay *x* / *M* =46 and 93 as well as beyond it for *x* / *M* ≈200. Here:

$$\Lambda\_u = \frac{\overline{u^{.2}} - \overline{u^{.2}\_0}}{\overline{u^{.2}\_0}}, \%, \quad \Lambda\_v = \frac{\overline{v^{.2}} - \overline{v^{.2}\_0}}{\overline{v^{.2}\_0}} \%, \quad \Lambda\_w = \frac{\overline{w^{.2}} - \overline{w^{.2}\_0}}{\overline{w^{.2}\_0}} \%. \tag{41}$$

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 *Δu*, *Δv* and *Δw* are uniform that corresponds to the initial grid-generated homogeneous isotropic turbulence, which

**Figure 5.** The calculated axial turbulence decay behind the grids: 1 and 2 are the data got for the unladen and parti‐ cle-laden flow, respectively, at *M*=4.8 mm, 3 and 4 are the data obtained for the same conditions at *M*=10 mm.

**Figure 4.** Axial turbulence decay behind the grid M=4.8 mm: 1 and 3 are the data [23] got for the unladen and parti‐

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55

cle-laden flows, respectively; 2 and 4 are the numerical data obtained for the same conditions.

**Figure 4.** Axial turbulence decay behind the grid M=4.8 mm: 1 and 3 are the data [23] got for the unladen and parti‐ cle-laden flows, respectively; 2 and 4 are the numerical data obtained for the same conditions.

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

**Figure 3.** Numerical and experimental [22] distributions of the normalized Reynolds normal stress tensor components.

was 9.5 m/s, the flow mass loading was 0.14 kg dust/kg air. The grids of the square mesh size *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

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

Figures 6 – 11 show the cross-section modifications of three components of the Reynolds stress, *Δu*, *Δv*, *Δw*, caused by 700-*μ*m glass beads, calculated by the presented RSTM model at two locations of the initial period of the grid-generated turbulence decay *x* / *M* =46 and 93 as well

> 2 2 2 2 2 2 0 0 0 2 2 2 0 0 0 ' ' ' ' ' ' ,%, %, %.


 ' ' *<sup>u</sup> <sup>v</sup> <sup>w</sup> u u v v w w u v w*

was used for the model validation. The mean flow velocity

of the physical density 2500 kg/m<sup>3</sup>

54 Computational and Numerical Simulations

as beyond it for *x* / *M* ≈200. Here:

'

for the grid *M*=4.8 mm.

described in [24].

**Figure 5.** The calculated axial turbulence decay behind the grids: 1 and 2 are the data got for the unladen and parti‐ cle-laden flow, respectively, at *M*=4.8 mm, 3 and 4 are the data obtained for the same conditions at *M*=10 mm.

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 *Δu*, *Δv* and *Δw* are uniform that corresponds to the initial grid-generated homogeneous isotropic turbulence, which decays downstream (s. Figures 4 and 5). The distributions of modification of *Δu*, *Δv* and *Δ<sup>w</sup>* remain uniform downstream. At the same time, the cross-section extent of uniformity of distributions of components of the Reynolds stress and the degree of the particles effect on turbulence decrease, since the turbulence level decreases downstream (cf. data presented for *x* / *M* =46 and 93 in Figures 6 – 11).

**Figure 7.** Effect of particles on the modification of the *y*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

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**Figure 8.** Effect of particles on the modification of the *z*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

**Figure 6.** Effect of particles on the modification of the *x*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

The distributions of modification of *Δu*, *Δv* and *Δw* taken place beyond the initial period of the turbulence decay (location *x* / *M* ≈200 at Figures 6 – 11) are typical of the channel turbulent particulate flow. One can see that in this case the turbulence enhancement becomes slower, since here the turbulence level is substantially smaller as compared with the initial period of decay, i.e. for *x* / *M* <100 (s. Figures 4 and 5). This means that the grid-generated turbulence of the particulate flow decays downstream, and this causes the decrease of the rate of turbulence enhancement due to the particles occurred beyond the initial period of the turbulence decay. As a result, the turbulence is attenuated, that is expressed in terms of decrease of *Δu* towards the pipe wall (s. Figure 9). Such tendency has been shown qualitatively in [25].

The certain increase of *Δu*, *Δv* and *Δw*, that is observed verge towards the wall (s. Figures 6 – 11), arises from the growth of the slip velocity (s. curves 1, 2, 3 in Figure 12). The decrease of *Δu*, *Δv* and *Δw* taken place in the immediate vicinity of the wall is caused by the decrease of the length scale of the energy-containing vortices and, thus, the increase of the dissipation of the turbulence kinetic energy.

decays downstream (s. Figures 4 and 5). The distributions of modification of *Δu*, *Δv* and *Δ<sup>w</sup>* remain uniform downstream. At the same time, the cross-section extent of uniformity of distributions of components of the Reynolds stress and the degree of the particles effect on turbulence decrease, since the turbulence level decreases downstream (cf. data presented for

**Figure 6.** Effect of particles on the modification of the *x*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

the pipe wall (s. Figure 9). Such tendency has been shown qualitatively in [25].

The distributions of modification of *Δu*, *Δv* and *Δw* taken place beyond the initial period of the turbulence decay (location *x* / *M* ≈200 at Figures 6 – 11) are typical of the channel turbulent particulate flow. One can see that in this case the turbulence enhancement becomes slower, since here the turbulence level is substantially smaller as compared with the initial period of decay, i.e. for *x* / *M* <100 (s. Figures 4 and 5). This means that the grid-generated turbulence of the particulate flow decays downstream, and this causes the decrease of the rate of turbulence enhancement due to the particles occurred beyond the initial period of the turbulence decay. As a result, the turbulence is attenuated, that is expressed in terms of decrease of *Δu* towards

The certain increase of *Δu*, *Δv* and *Δw*, that is observed verge towards the wall (s. Figures 6 – 11), arises from the growth of the slip velocity (s. curves 1, 2, 3 in Figure 12). The decrease of *Δu*, *Δv* and *Δw* taken place in the immediate vicinity of the wall is caused by the decrease of the length scale of the energy-containing vortices and, thus, the increase of the dissipation of

*x* / *M* =46 and 93 in Figures 6 – 11).

56 Computational and Numerical Simulations

the turbulence kinetic energy.

**Figure 7.** Effect of particles on the modification of the *y*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

**Figure 8.** Effect of particles on the modification of the *z*-normal component of the Reynolds stress: *M*=4.8 mm, *z*=0.

**Figure 9.** Effect of particles on the modification of the *x*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

**Figure 11.** Effect of particles on the modification of the *z*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

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**Figure 12.** The cross-section distributions of the axial gas and particles velocities and particles mass concentration for the grid *M*=4.8 mm: 1 –*u* /*U* for *x* / *M* =46; 2 –*u* /*U*, 3 –*us* / *U* and 4 –α / α*m* for *x* / *M* ≈200. Here α*m* is the value of the

mass concentration occurring at the flow axis.

**Figure 10.** Effect of particles on the modification of the *y*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

**Figure 11.** Effect of particles on the modification of the *z*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

**Figure 9.** Effect of particles on the modification of the *x*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

58 Computational and Numerical Simulations

**Figure 10.** Effect of particles on the modification of the *y*-normal component of the Reynolds stress: *M*=10 mm, *z*=0.

**Figure 12.** The cross-section distributions of the axial gas and particles velocities and particles mass concentration for the grid *M*=4.8 mm: 1 –*u* /*U* for *x* / *M* =46; 2 –*u* /*U*, 3 –*us* / *U* and 4 –α / α*m* for *x* / *M* ≈200. Here α*m* is the value of the mass concentration occurring at the flow axis.

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

The work was done within the frame of the target financing under the Project SF0140070s08 (Estonia) and supported by the ETF grant Project ETF9343 (Estonia). The authors are grateful for the technical support of Computational Biology Initiative High Performance Computing Center of University of Texas at San Antonio (USA) and Texas Advanced Computing Center in Austin (USA). This study is related to the activity of the European network action COST MP1106 "Smart and green interfaces - from single bubbles and drops to industrial, environ‐

, Medhat Hussainov1

1 Research Laboratory of Multiphase Media Physics, Faculty of Science, Tallinn University

2 School of Mechanical and Materials Engineering, Washington State University, Pullman,

[1] Elghobashi S.E., Abou-Arab T.W. A Two-Equation Turbulence Model for Two-Phase

[2] Pourahmadi F., Humphrey J.A.C. Modeling Solid-Fluid Turbulent Flows with Appli‐ cation to Predicting Erosive Wear. International Journal of Physicochemical Hydro‐

[3] Rizk M.A., Elghobashi S.E. A Two-Equation Turbulence Model for Dispersed Dilute Confined Two-Phase Flows. International Journal of Multiphase Flow 1989; 15(1)

, Igor Shcheglov1

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

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

61

, Sergei Tisler1

,

rate of the turbulence enhancement which is caused by particles.

**Acknowledgements**

**Author details**

Igor Krupenski1

Washington, USA

**References**

119-133.

mental and biomedical applications".

Alexander Kartushinsky1\*, Ylo Rudi1

of Technology, Tallinn, Estonia

and David Stock2

\*Address all correspondence to: aleksander.kartusinski@ttu.ee

Flows. Physics of Fluids 1983; 26(4) 931-938.

dynamics 1983; 4(3) 191-219.

**Figure 13.** Effect of particles on the modification of the turbulence kinetic energy: 1 –*M* =4.8mm,*x* / *M* =46; 2 – *M* =4.8mm,*x* / *M* =93; 3 –*M* =10mm,*x* / *M* =46; 4 –*M* =10mm,*x* / *M* =93.

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.
